# Class 12 Maths NCERT Solutions for Chapter 5 Continuity and Differentiability Exercise 5.1

### Continuity and Differentiability Exercise 5.1 Solutions

**1. Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.**

**Solution**

The given function is f(x) = 5x - 3

At x = 0, f(0) = 5× 0 -3 = -3

Therefore, f is continuous at x = 5

**2. Examine the continuity of the function f (x) = 2x ^{2} – 1 at x = 3.**

**Solution**

The given function is f(x) = 2x^{2} - 1

At x = 3, f(x) = f(3) = 2× 3^{2} - 1 = 17

Thus, f is continuous at x = 3

**3. Examine the following functions for continuity. (i) f(x) = x - 5 (ii) f(x) = [1/(x- 5)] , x ≠ 5 (iii) f(x) = (x**

^{2}- 25)/(x + 5), x ≠ - 5 (iv) f(x) = |x - 5|, x ≠ 5

**Solution**

(i) The given function is f(x) = x - 5

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

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

(ii) The given function is f(x) = [1/(x- 5)] , x ≠ 5

For any real number k ≠ 5, we obtain

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

(iii) The given function is f(x) = (x^{2} - 25)/(x + 5), x ≠ - 5

For any real number c ≠ - 5 , we obtain

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

(iv) The given function is

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, f(c) = 5 - c

Therefore, f is continuous at all real numbers less than 5. **Case II :** c = 5

Then, f(c) = f(5) = (5 - 5) = 0

Therefore, f is continuous at x = 5 **Case III : **c > 5

Then, f(c) = f(5) = c - 5

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

**4. Prove that the function f(x) = x**

^{n}is continuous at x = n, where n is a positive integer.**Solution**

^{n}

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

^{n}.

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

**5. Is the function**

*f*defined by f(x)=**continuous at**

*x*= 0? At*x*= 1? At*x*= 2?**Solution**

At x = 0,

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

Therefore, f is continuous at x = 0

At x = 1,

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

The left hand limit of f at x = 1 is,

Therefore, f is not continuous at x = 1

At x = 2,

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

Therefore, f is continuous at x = 2

**6. Find all points of discontinuity of f, where f is defined by**

**Solution**

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

c < 2

c > 2

c = 2

**Case I :**c < 2

f(c) = 2c + 3

Then,

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

**Case II :**c > 2

Then,

f(c) = 2c - 3

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

**Case III :**c = 2

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

Therefore, f is not continuous at x = 2 .

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

**7. Find all points of discontinuity of f, where f is defined by**

**Solution**

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:**

If c < -3, then f(c) = -c + 3

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

**Case II:**

If c = -3, then f(-3) = -(-3) + 3 = 6

Therefore, f is continuous at x = -3

**Case III :**

If -3 < c <3, then f(c)

**Case IV :**

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

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 :**

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

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

**8. Find all points of discontinuity of f, where f is defined by**

**Solution**

It is known that, x < 0 ⇒ |x| = -x and x > 0 ⇒ |x| = x

Therefore, the given function can be rewritten as

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:**

If c < 0, then f(c) = -1

Therefore, f is continuous at all points x < 0

**Case II :**

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

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 :**

If c > 0, then f(c) = 1

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

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

**9. Find all points of discontinuity of**

*f*, where*f*is defined by**Solution**

It is known that, x < 0 ⇒ |x| = -x

Therefore, the given function can be rewritten as

Hence, the given function has no point of discontinuity.

**10. Find all points of discontinuity of**

*f*, where*f*is defined by**Solution**

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 :**

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

**Case II :**

If c = 1, then f(c) = f(1) = 1 + 1 = 2

The left hand limit of f at x = 1 is,

Hence, the given function f has no point of discontinuity.

**11. Find all points of discontinuity of f, where f is defined**

**Solution**

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

Let c be a point on the real line.

**Case I:**

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

Hence, f has no point of discontinuity.

**12.Find all points of discontinuity of f, where f is defined by f(x)=**

**Solution**

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:**

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,

Therefore, f is not continuous at x = 1 .

**Case III :**

If c > 1, then f(c) = c

^{2}

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.

**13. Is the function defined by**

**a continuous function ?**

**Solution**

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:**

**Case II :**

If c = 1, then f(1) = 1 + 5 = 6

The left hand limit of f at x = 1 is,

Therefore, f is not continuous at x = 1

**Case III :**

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

**14. Discuss the continuity of the function f, where f is defined by**

**Solution**

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 :**

**Case II :**

If c = 1, then f(3) = 3

The left hand limit of f at x = 1 is,

Therefore, f is not continuous at x = 1

**Case III :**

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

**Case IV :**

If c = 3, then f(c) = 5

The left hand limit of f at x = 3 is,

Therefore, f is not continuous at x = 3

**Case V :**

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

Hence, f is not continuous at x = 1 and x = 3

**15. Discuss the continuity of the function**

*f*, where*f*is defined by**Solution**

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

Let c be a point on the real line.

**Case I :**

If c < 0 , then f(c) = 2c

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

**Case II :**

If c = 0, then f(c) = f(0) = 0

The left hand limit of f at x = 0 is

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

**Case IV :**

If c = 1, then f(c) = f(1) = 0

The left hand limit of f at x = 1 is,

Therefore, f is not continuous at x = 1

**Case V :**

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

Hence, f is not continuous only at x = 1

**16. Discuss the continuity of the function**

*f*, where*f*is defined by**Solution**

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

Let c be a point on the real line.

**Case I :**

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

**Case II :**

If c = -1, then f(c) = f(-1) = -2

The left hand limit of f at x = -1 is ,

Therefore, f is continuous at x = -1

**Case III :**

If - < c < 1, then f(c) = 2c

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

**Case IV :**

If c = 1, then f(c) = f(1) = 2 × 1 = 2

The left hand limit of f at x = 1 is,

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.

**17. Find the relationship between**

*a*and*b*so that the function*f*defined by**is continuous at x = 3.**

**Solution**

Therefore, from (1), we obtain

3a + 1 = 3b + 3

⇒ 3a + 1 = 3b + 3

⇒ 3a = 3b = 2

⇒a = b + 2/3

Therefore, the required relationship is given by , a = b + 2/3

**18. For what value of Î» is the function defined by**

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

**Solution**

If f is continuous at x = 0, then

⇒ Î»(0

^{2}– 2 × 0) = 4 × 0 + 1 = 0

⇒ 0 = 1 = 0, which is not possible

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

At x = 1,

f(1) = 4x + 1 = 4 × 1 + 1 = 5

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

**19. Show that the function defined by g(x) = x = [x] is discontinuous at all integral point. Here [x] denotes the greatest integer less than or equal to**

*x*.**Solution**

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

Let n be an integer.

Then,

g(n) = n - [n] = n - n = 0

The left hand limit of f at x = n is,

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.

**20. Is the function defined by f(x) = x**

^{2}- sinx + 5 continuous at x = Ï€ ?**Solution**

^{2}- sinx + 5

It is evident that f is defined at x = Ï€

At x = Ï€, f(x) = f(Ï€) = Ï€

^{2}- sinÏ€ + 5 = Ï€

^{2}- 0 + 5 = Ï€

^{2}+ 5

Therefore, the given function f is continuous at x = Ï€

**21. Discuss the continuity of the following functions.**

(a)

(a)

*f*(*x*) = sin*x*+ cos*x*

(b)*f*(*x*) = sin*x*− cos*x*

(c)*f*(*x*) = sin*x*× cos x**Solution**

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

Let g(x) = sinx

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

Let c be a real number. Put x = c + h

If x → c, then h → 0

g(c) = sinc

Let h (x) = cos x

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

Let C be a real number. Put x = c + h

If x → c, then h → 0

h(c) = cos c

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

**22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions,**

**Solution**

(i) h(x)/g(x) , g(x) ≠ 0 is continuous

(ii) 1/g(x), g(x) ≠ 0 is continuous

(iii) 1/h(x), h(x) ≠ 0 is continuous

Let g(x) = sin x

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

Let c be a real number. Put x = (c + h)

if x → c, then h → 0

g(c) = sin c

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

If x → c, then h → 0

h(c) = cos c

cosec x = 1/sin x, sin x ≠ 0 is continuous

⇒ cosec x, x ≠ nÏ€ (n ∊ Z) is continuous

Therefore, cosecant if continuous except at x = np, n ∊ Z

sec x = 1/cos x, cos x ≠ 0 is continuous

⇒ sec x, x ≠ (2n + 1)(Ï€/2) (n ∊ Z) is continuous

cot x = cos x/sin x , sin x ≠ 0 is continuous

**23. Find the points of discontinuity of f, where**

**Solution**

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

**Case I :**

**Case III :**

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

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

**24. Determine if f defined by**

**is a continuous function ?**

**Solution**

**Case I :**

^{2}sin 1/c

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

**25. Examine the continuity of f, where f is defined b**

**Solution**

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

**Case I :**

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

**Case II :**

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

**26. Find the values of**

*k*so that the function*f*is continuous at the indicated point.**at x = Ï€/2**

**Solution**

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

⇒ k = 6

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

**at x = 2**

**Solution**

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

^{2}= 4k

⇒ k × 2

^{2}= 3 = 4k

⇒ 4k = 3

⇒ k = 3/4

Therefore, the required value of k is 3/4.

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

**at x = Ï€**

**Solution**

⇒ kÏ€ + 1 = -1 = kÏ€ + 1

⇒ k = -2/Ï€

Therefore, the required value of k is -2/Ï€.

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

**at x = 5**

**Solution**

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 f is defined at x = 5 and f(5) = kx + 1 = 5k + 1

⇒ 5k + 1 = 15 - 5 = 5k + 1

⇒ 5k + 1 = 10

⇒ 5k = 9

⇒ k = 9/5

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

**is a continuous function.**

**Solution**

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

Since f is continuous at x = 2, we obtain

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

⇒ a = 2

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

⇒ 4 + b = 5

⇒ b = 1

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

**31. Show that the function defined by f(x) = cos (x**

^{2}) is a continuous function .**Solution**

^{2})

^{2}

**[∵ (goh)(x) = g[h(x)] = g(x**

^{2}) = cos(x^{2}) = f(x) ]^{2}are continuous functions.

If x ⟶ c, then h ⟶ 0

^{2}

Clearly, h is defined for every real number.

^{2}

Therefore, h is a continuous function.

^{2}) is a continuous function.

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

**Solution**

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

**[∵ (goh) (x) = g{h (x)} = g(cos x) = |cos x| = f(x)]**

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

Clearly, g is defined d for all real numbers.

**Case I :**

**Case II :**

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

**Case III :**

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

If x → c, then h → 0

h(c) = cos c

**33. Examine sin |x| is a continuous function.**

**Solution**

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

**[ ∵ (goh)(x) = {h(x)} = g(sin x) = |sin x| = f(x)]**

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

g(x) = |x| can be written as

Clearly, g is defined for all real numbers.

**Case I :**

**Case II :**

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

**Case III :**

Therefore, g is continuous at x = 0

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

h(x) = sin x

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

If x → c, then k→ 0

h (c) = sin c

Therefore, h is a continuous function.

**34. Find all the points of discontinuity of f defined by f(x) = |x| - |x + 1|.**

**Solution**

The two functions, g and h, are defined as

Then, f = g - h

The continuity of g and h is examined first.

Clearly, g is defined for all real numbers.

**Case I :**

**Case II :**

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

**Case III :**

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

h(x) = |x + 1| can be written as

**Case I :**

**Case II :**

**Case III :**

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.