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Exercise 5.1 - Chapter 5 Continuity & Differentiability class 12 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Prove that the functionhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_1e328cf0.gifis continuous athttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_48cc118a.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_a69856a.gif

Therefore, f is continuous at x = 0

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_m3baa23bc.gif

Therefore, is continuous at x = −3

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_75094717.gif

Therefore, f is continuous at x = 5

Question 2:

Examine the continuity of the functionhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6751/Chapter%205_html_m13c6d82c.gif.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6751/Chapter%205_html_266c3190.gif

Thus, f is continuous at x = 3

Question 3:

Examine the following functions for continuity.

(a)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m63a64d7c.gif (b)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m676cbc31.gif

(c) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m7502a14e.gif (d) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m2bcad135.gif

Answer:

(a) The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m63a64d7c.gif

It is evident that f is defined at every real number k and its value at k is k − 5.

It is also observed that, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m4a84a954.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_13958887.gif

Hence, f is continuous at every real number and therefore, it is a continuous function.

(b) The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m676cbc31.gif

For any real number k ≠ 5, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_50a3ea5c.gif

Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function.

(c) The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m7502a14e.gif

For any real number c ≠ −5, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m3f3d37a6.gif

Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function.

(d) The given function is https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_2c77b583.gif

This function f is defined at all points of the real line.

Let c be a point on a real line. Then, c < 5 or c = 5 or c > 5

Case I: c < 5

Then, (c) = 5 − c

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m1296b13d.gif

Therefore, f is continuous at all real numbers less than 5.

Case II : c = 5

Then, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m37a44f8.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_c8fc2c0.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_15e897ad.gif

Therefore, is continuous at x = 5

Case III: c > 5

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m45cd474e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous at all real numbers greater than 5.

Hence, f is continuous at every real number and therefore, it is a continuous function.

Question 4:

Prove that the function https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6755/Chapter%205_html_m7cd8b8a0.gifis continuous at x = n, where n is a positive integer.

Answer:

The given function is f (x) = xn

It is evident that f is defined at all positive integers, n, and its value at n is nn.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6755/Chapter%205_html_75e17cf3.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6755/Chapter%205_html_4dc28689.gif

Therefore, is continuous at n, where n is a positive integer.

Question 5:

Is the function f defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_m43306763.gif

continuous at x = 0? At x = 1? At x = 2?

Answer:

The given function f is https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_m43306763.gif

At x = 0,

It is evident that f is defined at 0 and its value at 0 is 0.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_1a9d0cd6.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_m5835d82a.gif

Therefore, f is continuous at x = 0

At x = 1,

is defined at 1 and its value at 1 is 1.

The left hand limit of f at x = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_58751911.gif

The right hand limit of at x = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_m12aff82a.gif

Therefore, f is not continuous at x = 1

At = 2,

is defined at 2 and its value at 2 is 5.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_2480fbf6.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_189a3d61.gif

Therefore, f is continuous at = 2

Question 6:

Find all points of discontinuity of f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_6f5d7057.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_6f5d7057.gif

It is evident that the given function f is defined at all the points of the real line.

Let c be a point on the real line. Then, three cases arise.

(i) c < 2

(ii) c > 2

(iii) c = 2

Case (i) c < 2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_m78776ee0.gif

Therefore, f is continuous at all points x, such that x < 2

Case (ii) c > 2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_m5ef9a613.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous at all points x, such that x > 2

Case (iii) c = 2

Then, the left hand limit of at x = 2 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_mbe21158.gif

The right hand limit of f at x = 2 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_6855250.gif

It is observed that the left and right hand limit of f at x = 2 do not coincide.

Therefore, f is not continuous at x = 2

Hence, x = 2 is the only point of discontinuity of f.

Question 7:

Find all points of discontinuity of f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_m763140dc.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_m66ddd76d.gif

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_67637c17.gif

Therefore, f is continuous at all points x, such that x < −3

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_6d71b1df.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_727680e3.gif

Therefore, f is continuous at x = −3

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_m2903fe5e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous in (−3, 3).

Case IV:

If c = 3, then the left hand limit of at x = 3 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_5941cb89.gif

The right hand limit of at x = 3 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_29f0b36.gif

It is observed that the left and right hand limit of f at x = 3 do not coincide.

Therefore, f is not continuous at x = 3

Case V:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_79947981.gif

Therefore, f is continuous at all points x, such that x > 3

Hence, x = 3 is the only point of discontinuity of f.

Question 8:

Find all points of discontinuity of f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m4ccee9e7.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m4ccee9e7.gif

It is known that,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_5509af4.gif

Therefore, the given function can be rewritten as

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m292cde6.gif

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m28f79be1.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous at all points x < 0

Case II:

If c = 0, then the left hand limit of at x = 0 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_3fa92674.gif

The right hand limit of at x = 0 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m4c329f7d.gif

It is observed that the left and right hand limit of f at x = 0 do not coincide.

Therefore, f is not continuous at x = 0

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m4bef27dd.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous at all points x, such that x > 0

Hence, x = 0 is the only point of discontinuity of f.

Question 9:

Find all points of discontinuity of f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_m3ec67b4a.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_m3ec67b4a.gif

It is known that,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_m27c7ceea.gif

Therefore, the given function can be rewritten as

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_2ac3ed3e.gif

Let c be any real number. Then, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_3ef9fc07.gif

Also,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_378e5a68.gif

Therefore, the given function is a continuous function.

Hence, the given function has no point of discontinuity.

Question 10:

Find all points of discontinuity of f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m5cd13fd3.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m5cd13fd3.gif

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_3501b6a4.gif

Therefore, f is continuous at all points x, such that x < 1

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m4ba2db85.gif

The left hand limit of at x = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_4d9c2e78.gif

The right hand limit of at x = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m2b97949e.gif

Therefore, f is continuous at x = 1

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m68ca6455.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous at all points x, such that x > 1

Hence, the given function has no point of discontinuity.

Question 11:

Find all points of discontinuity of f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_224340ea.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_224340ea.gif

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_m35d1b7b9.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous at all points x, such that x < 2

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_m442e0a43.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_6003b281.gif

Therefore, f is continuous at x = 2

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_m2052425d.gif

Therefore, f is continuous at all points x, such that x > 2

Thus, the given function f is continuous at every point on the real line.

Hence, has no point of discontinuity.

Question 12:

Find all points of discontinuity of f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_21ccc991.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_21ccc991.gif

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_2b68e9d.gif

Therefore, f is continuous at all points x, such that x < 1

Case II:

If c = 1, then the left hand limit of f at x = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_m4a164abb.gif

The right hand limit of f at = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_6e433823.gif

It is observed that the left and right hand limit of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_6d5b96ba.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.

Question 13:

Is the function defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m1d76af7c.gif

a continuous function?

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m1d76af7c.gif

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_71e1a3a3.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous at all points x, such that x < 1

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m49f14e3.gif

The left hand limit of at x = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m33dba451.gif

The right hand limit of f at = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_47b80a2.gif

It is observed that the left and right hand limit of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_3afae62a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m3078a2af.gif

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.

Question 14:

Discuss the continuity of the function f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_20fb2c5f.gif

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_20fb2c5f.gif

The given function is defined at all points of the interval [0, 10].

Let c be a point in the interval [0, 10].

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m682944c5.gif

Therefore, f is continuous in the interval [0, 1).

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m6367ab7.gif

The left hand limit of at x = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m7a7b2037.gif

The right hand limit of f at = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_49b439a6.gif

It is observed that the left and right hand limits of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m6269866a.gif

Therefore, f is continuous at all points of the interval (1, 3).

Case IV:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_me0f6680.gif

The left hand limit of at x = 3 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m7ac0d075.gif

The right hand limit of f at = 3 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m590ecce0.gif

It is observed that the left and right hand limits of f at x = 3 do not coincide.

Therefore, f is not continuous at x = 3

Case V:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m561a1438.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_5a589017.gif

Therefore, f is continuous at all points of the interval (3, 10].

Hence, is not continuous at = 1 and = 3

Question 15:

Discuss the continuity of the function f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_m24dbf833.gif

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_m24dbf833.gif

The given function is defined at all points of the real line.

Let c be a point on the real line.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_964708f.gif

Therefore, f is continuous at all points x, such that x < 0

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_4fe26b04.gif

The left hand limit of at x = 0 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_15c64d62.gif

The right hand limit of f at = 0 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_6c4c75cf.gif

Therefore, f is continuous at x = 0

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_m18bc350f.gif

Therefore, f is continuous at all points of the interval (0, 1).

Case IV:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_6b889dd5.gif

The left hand limit of at x = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_d825cb9.gif

The right hand limit of f at = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_30c158c5.gif

It is observed that the left and right hand limits of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case V:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_m2bca5b0a.gif

Therefore, f is continuous at all points x, such that x > 1

Hence, is not continuous only at = 1

Question 16:

Discuss the continuity of the function f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_5a22c2c0.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_5a22c2c0.gif

The given function is defined at all points of the real line.

Let c be a point on the real line.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_2d706456.gif

Therefore, f is continuous at all points x, such that x < −1

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m5c4dc14e.gif

The left hand limit of at x = −1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_71dd50a6.gif

The right hand limit of f at = −1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_7e7fb68a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m2c8a5e24.gif

Therefore, f is continuous at x = −1

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m1f17b7c9.gif

Therefore, f is continuous at all points of the interval (−1, 1).

Case IV:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m441906ff.gif

The left hand limit of at x = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m6ed05fd3.gif

The right hand limit of f at = 1 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_518dec0f.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_2124cbf9.gif

Therefore, f is continuous at x = 2

Case V:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m56801209.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_5a589017.gif

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observations, it can be concluded that f is continuous at all points of the real line.

Question 17:

Find the relationship between a and b so that the function f defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_7394646a.gif

is continuous at = 3.

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_7394646a.gif

If f is continuous at x = 3, then

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_m33fde032.gif

Therefore, from (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_4ee69190.gif

Therefore, the required relationship is given by,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_m11cd780a.gif

Question 18:

For what value of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_m11cc021f.gifis the function defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_2ac5cb7a.gif

continuous at x = 0? What about continuity at x = 1?

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_2ac5cb7a.gif

If f is continuous at x = 0, then

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_5b2103db.gif

Therefore, there is no value of λ for which f is continuous at x = 0

At x = 1,

f (1) = 4x + 1 = 4 × 1 + 1 = 5

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_m7a9447a8.gif

Therefore, for any values of λ, f is continuous at x = 1

Question 19:

Show that the function defined by https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m40f6c85a.gifis discontinuous at all integral point. Here https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m2694cf9.gifdenotes the greatest integer less than or equal to x.

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m40f6c85a.gif

It is evident that g is defined at all integral points.

Let n be an integer.

Then,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m207aa0cb.gif

The left hand limit of at x = n is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m1c2be5f0.gif

The right hand limit of f at n is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m172b5fc1.gif

It is observed that the left and right hand limits of f at x = n do not coincide.

Therefore, f is not continuous at x = n

Hence, g is discontinuous at all integral points.

Question 20:

Is the function defined by https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6792/Item%2020_html_m3add3190.gifcontinuous at =

π?

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6792/Item%2020_html_m3add3190.gif

It is evident that f is defined at =

π.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6792/Item%2020_html_m7c663ad3.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6792/Item%2020_html_m588c68e3.gif

Therefore, the given function f is continuous at = π

Question 21:

Discuss the continuity of the following functions.

(a) f (x) = sin x + cos x

(b) f (x) = sin x − cos x

(c) f (x) = sin x × cos x

Answer:

It is known that if and are two continuous functions, then

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6796/Chapter%205_html_m60aee736.gifare also continuous.

It has to proved first that g (x) = sin and h (x) = cos x are continuous functions.

Let (x) = sin x

It is evident that g (x) = sin x is defined for every real number.

Let be a real number. Put x = c + h

If x → c, then h → 0

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6796/Chapter%205_html_m40481c7a.gif

Therefore, g is a continuous function.

Let h (x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let be a real number. Put x = c + h

If x → c, then h → 0

(c) = cos c

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6796/Chapter%205_html_m60ec81a5.gif

Therefore, h is a continuous function.

Therefore, it can be concluded that

(a) f (x) = g (x) + h (x) = sin x + cos x is a continuous function

(b) f (x) = g (x) − h (x) = sin x − cos x is a continuous function

(c) f (x) = g (x) × h (x) = sin x × cos x is a continuous function

Question 22:

Discuss the continuity of the cosine, cosecant, secant and cotangent functions,

Answer:

It is known that if and are two continuous functions, then

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_b9dd9b.gif

It has to be proved first that g (x) = sin and h (x) = cos x are continuous functions.

Let (x) = sin x

It is evident that g (x) = sin x is defined for every real number.

Let be a real number. Put x = c + h

If x

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m48851eb.gif c, then h

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m48851eb.gif0

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m40481c7a.gif

Therefore, g is a continuous function.

Let h (x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let be a real number. Put x = c + h

If x ® c, then h ® 0

(c) = cos c

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m60ec81a5.gif

Therefore, h (x) = cos x is continuous function.

It can be concluded that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m6892dbfe.gif

Therefore, cosecant is continuous except at np, Î Z

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_mb65bf10.gif

Therefore, secant is continuous except at https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m1aadeded.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_4fef99da.gif

Therefore, cotangent is continuous except at np, Î Z

Question 23:

Find the points of discontinuity of f, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_m324e9506.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_m324e9506.gif

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_m7f62ef9c.gif

Therefore, f is continuous at all points x, such that x < 0

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_30eff037.gif

Therefore, f is continuous at all points x, such that x > 0

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_2b68db9d.gif

The left hand limit of f at x = 0 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_m7ab148d0.gif

The right hand limit of f at x = 0 is,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_mae478ee.gif

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at all points of the real line.

Thus, f has no point of discontinuity.

Question 24:

Determine if f defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_1bf52e85.gif

is a continuous function?

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_1bf52e85.gif

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_5690438c.gif

Therefore, f is continuous at all points ≠ 0

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_73e8f29.gif https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Selection_007(10).png

⇒-x2≤x2sin1x≤x2https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Selection_009(12).png https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_m473a2a2d.gif

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at every point of the real line.

Thus, f is a continuous function.

Question 25:

Examine the continuity of f, where f is defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_75920e75.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_75920e75.gif

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_31cd584e.gif

Therefore, f is continuous at all points x, such that x ≠ 0

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_m14ad73a7.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_c367a60.gif

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at every point of the real line.

Thus, f is a continuous function.

Question 26:

Find the values of so that the function f is continuous at the indicated point.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_50475cb3.gif

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_338e09a5.gif

The given function f is continuous athttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gif, if f is defined at https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gifand if the value of the f at https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gif equals the limit of f athttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gif.

It is evident that is defined athttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gif andhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m46f41e75.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_46e34999.gif

Therefore, the required value of k is 6.

Question 27:

Find the values of so that the function f is continuous at the indicated point.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_m2d83caea.gif

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_38fec36d.gif

The given function f is continuous at x = 2, if f is defined at x = 2 and if the value of f at x = 2 equals the limit of f at x = 2

It is evident that is defined at x = 2 andhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_a656feb.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_79a9c467.gif

Therefore, the required value ofhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_m78d73418.gif.

Question 28:

Find the values of so that the function f is continuous at the indicated point.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_66eb254b.gif

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_12f6ac2f.gif

The given function f is continuous at x = p, if f is defined at x = p and if the value of f at x = p equals the limit of f at x = p

It is evident that is defined at x = p andhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_m501255cb.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_m1ce988b8.gif

Therefore, the required value ofhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_m563bdc98.gif

Question 29:

Find the values of so that the function f is continuous at the indicated point.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_1b0651b3.gif

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_m71365e67.gif

The given function f is continuous at x = 5, if f is defined at x = 5 and if the value of f at x = 5 equals the limit of f at x = 5

It is evident that is defined at x = 5 andhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_a7f4fd3.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_19cdbff6.gif

Therefore, the required value ofhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_m5e28c8c3.gif

 

Question 30:

Find the values of a and b such that the function defined by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6833/Chapter%205_html_m49453141.gif

is a continuous function.

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6833/Chapter%205_html_m49453141.gif

It is evident that the given function f is defined at all points of the real line.

If f is a continuous function, then f is continuous at all real numbers.

In particular, f is continuous at = 2 and = 10

Since f is continuous at = 2, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6833/Chapter%205_html_m6a67a506.gif

Since f is continuous at = 10, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6833/Chapter%205_html_m62db6148.gif

On subtracting equation (1) from equation (2), we obtain

8a = 16

⇒ a = 2

By putting a = 2 in equation (1), we obtain

2 × 2 + b = 5

⇒ 4 + b = 5

⇒ b = 1

Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively.

 

Question 31:

Show that the function defined by f (x) = cos (x2) is a continuous function.

Answer:

The given function is (x) = cos (x2)

This function f is defined for every real number and f can be written as the composition of two functions as,

f = g o h, where g (x) = cos x and h (x) = x2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6835/Chapter%205_html_2414acbb.gif

It has to be first proved that (x) = cos x and h (x) = x2 are continuous functions.

It is evident that g is defined for every real number.

Let c be a real number.

Then, g (c) = cos c

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6835/Chapter%205_html_m6d0b3abd.gif

Therefore, g (x) = cos x is continuous function.

h (x) = x2

Clearly, h is defined for every real number.

Let k be a real number, then h (k) = k2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6835/Chapter%205_html_m129f05e7.gif

Therefore, h is a continuous function.

It is known that for real valued functions and h,such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.

Therefore, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6835/Chapter%205_html_71df85f5.gifis a continuous function.

Question 32:

Show that the function defined byhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m67314a85.gif is a continuous function.

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m67314a85.gif

This function f is defined for every real number and f can be written as the composition of two functions as,

f = g o h, wherehttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m252b46c6.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m400b359b.gif

It has to be first proved that https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m252b46c6.gif are continuous functions.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_5c7c3891.gif

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m28e427f3.gif

Therefore, g is continuous at all points x, such that x < 0

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_7eaf7bf6.gif

Therefore, g is continuous at all points x, such that x > 0

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m40b8e9ef.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_7f384cbb.gif

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points.

(x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let be a real number. Put x = c + h

If x → c, then h → 0

(c) = cos c

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m60ec81a5.gif

Therefore, h (x) = cos x is a continuous function.

It is known that for real valued functions and h,such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.

Therefore, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m3735b0cc.gifis a continuous function.

Question 33:

Examine that https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_266e30fc.gif is a continuous function.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_4c157e2.gif

This function f is defined for every real number and f can be written as the composition of two functions as,

f = g o h, wherehttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m404b820b.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_4d6be309.gif

It has to be proved first that https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m404b820b.gif are continuous functions.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_5c7c3891.gif

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m28e427f3.gif

Therefore, g is continuous at all points x, such that x < 0

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_7eaf7bf6.gif

Therefore, g is continuous at all points x, such that x > 0

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m40b8e9ef.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_7f384cbb.gif

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points.

(x) = sin x

It is evident that h (x) = sin x is defined for every real number.

Let be a real number. Put x = c + k

If x → c, then k → 0

(c) = sin c

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_4d11afc.gif

Therefore, h is a continuous function.

It is known that for real valued functions and h,such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.

Therefore, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m6ee2e49d.gifis a continuous function.

Question 34:

Find all the points of discontinuity of defined byhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_163ea7b7.gif.

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_163ea7b7.gif

The two functions, g and h, are defined as

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m5b88ece7.gif

Then, f = − h

The continuity of g and is examined first.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_5c7c3891.gif

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m28e427f3.gif

Therefore, g is continuous at all points x, such that x < 0

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_7eaf7bf6.gif

Therefore, g is continuous at all points x, such that x > 0

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m40b8e9ef.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_7f384cbb.gif

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_2697ef2d.gif

Clearly, h is defined for every real number.

Let be a real number.

Case I:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_602d838d.gif

Therefore, h is continuous at all points x, such that x < −1

Case II:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m7a5a8238.gif

Therefore, h is continuous at all points x, such that x > −1

Case III:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_a45610c.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m56c2d36f.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m2f351985.gif

Therefore, h is continuous at x = −1

From the above three observations, it can be concluded that h is continuous at all points of the real line.

g and h are continuous functions. Therefore, g − is also a continuous function.

Therefore, has no point of discontinuity.

Also Read : Exercise-4.1-Chapter-4-Determinants-class-12-ncert-solutions-Maths

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