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

Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability:- Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability introduces students to the fundamental concepts of continuity and differentiability in functions. This exercise focuses on understanding when and how a function is continuous, ensuring there are no abrupt changes in its value, and differentiable, meaning it has a defined tangent at every point. 

By exploring these concepts, students gain a deeper insight into the behaviour of functions, which is crucial for advanced calculus and real-world applications. The exercise includes a variety of problems designed to reinforce these principles through practical examples and theorems. Check out the Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability NCERT Solutions from the below article. 

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

Along with the Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability NCERT Solutions, check out the other Class 12 Maths Chapter 5 Exercise solutions:-

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

Solution :

Question 2. Examine the continuity of the function f(x) = 2x2 – 1 at x = 3

Solution :

Thus, f is continuous at x = 3

Question 3. Examine the following functions for continuity.

(a)

(c)

Solution :

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 f(x) = xn is continuous at x = n, where n is a positive integer.

Solution :
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.

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

Question 5. Is the function f defined by

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

Solution :
The given function f is

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,

The right 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

Check Out: NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.2

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

Solution :

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

Solution :
The given function f is

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 < −3

Case II:

Therefore, f is continuous at x = −3

Case III:

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

Case IV:

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

The right 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.

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

Solution :

Check out: NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3

Question 9. Find all points of discontinuity of f, where f is defined by

Solution :

Question 10. Find all points of discontinuity of f, where f is defined by

Solution :

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

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

Check out: NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.5

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

Solution :

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, f has no point of discontinuity.

Check out: NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.6

Question 12. Find all points of discontinuity of f, where f is defined by

Solution :
The given function f is

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

Let c be a point on the real line.

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  a continuous function?

Solution :
The given function is

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

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

NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.8

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

f =

Solution :
The given function is f =

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:

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

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

Question 15. Discuss the continuity of the function f, where f is defined by

Solution :
The given function is

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

Let c be a point on the real line.

Case I:

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Question 16. Discuss the continuity of the function f, where f is defined by

Solution :
The given function f is

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

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 

is continuous at x = 3.

Solution :
The given function f is

If f is continuous at x = 3, then

Question 18. For what value of λ is the function defined by 

continuous at x = 0?

What about continuity at x = 1?

Solution :
The given function f is

If f is continuous at x = 0, then

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

Question 19. Show that the function defined by is discontinuous at all integral point. Here [denotes the greatest integer less than or equal to x. 

Solution :
The given function is

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

Let n be an integer.

Then,

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 continuous at x = π ?

Solution :
The given function is

It is evident that f is defined at x = π

Therefore, the given function f is continuous at x = 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 

Solution :
It is known that if g and h are two continuous functions, then

g + h, g – h and g.h  are also continuous.

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

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

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

If x → c, then h → 0

h (c) = cos c

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,

Solution :
It is known that if g and h are two continuous functions, then

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

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

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

If x → c, then h → 0

h (c) = cos c

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

It can be concluded that,

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

Solution :
The given function f is

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

Let c be a real number.

Case I:

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  is a continuous function?

Solution :
The given function f is

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

Let c be a real number.

Case I:

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

Solution :
The given function f is

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

Let c be a real number.

Case I:

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 k so that the function f is continuous at the indicated point.

Solution :
The given function f is

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

It is evident that f is defined at x = π/2 and f( π/2) = 3

Therefore, the required value of k is 6.

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

Solution :
The given function is

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 f is defined at x = 2 and f(2) = k(2)2 = 4k

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

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

Solution :
The given function is

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 f is defined at x = p and f(π) = kπ + 1

Therefore, the required value of k is -2/π

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

Solution :
The given function f is

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

Therefore, the required value of k is 9/5

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

is a continuous function. 

Solution :
The given function f is

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 x = 2 and x = 10

Since f is continuous at x = 2, we obtain

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.

Solution :
The given function is f (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

It has to be first proved that g (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

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

Therefore, h is a continuous function.

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

Therefore, h is a continuous function.

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

Solution :
The given function is f(x) = |cos x|

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) = |x| and h(x) = cos x

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

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

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

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

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

Therefore, is a continuous function.

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Question 33. Examine that sin|x| is a continuous function.

Solution :
Let, f(x) = sin|x|

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) = |x| and h (x) = sin x

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

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

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

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.

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

If x → c, then k → 0

h (c) = sin c

Therefore, h is a continuous function.

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

Therefore, is a continuous function.

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

Solution :
The given function is f(x) = |x| – |x + 1|

The two functions, g and h, are defined as

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, f = g − h is also a continuous function.

Therefore, f has no point of discontinuity.

Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability Summary

This exercise introduces the important concepts of continuity and differentiability in functions. It starts by explaining what it means for a function to be continuous at a point and over an interval, highlighting the necessary conditions for continuity. It also covers differentiability, showing that a function is differentiable at a point if it has a well-defined tangent. The exercise also emphasises that a differentiable function is always continuous.

Continuity

• Definition and conditions for a function to be continuous.

• Methods to check the continuity of functions at points and over intervals.

• Continuity in composite functions.

Differentiability

• Definition and conditions for differentiability.

• The relationship between continuity and differentiability.

• Differentiability in composite functions.

Important Theorems

• Intermediate Value Theorem.

• Rolle’s Theorem.

• Mean Value Theorem.

Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability FAQs

Q1. What is the definition of continuity at a point?

Ans. A function f(x)f(x)f(x) is continuous at a point x=cx = cx=c if lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c)limx→c​f(x)=f(c).

Q2. What are the conditions for a function to be continuous?

Ans. A function is continuous at a point x=cx = cx=c if:

1. f(c)f(c)f(c) is defined.

2. lim⁡x→cf(x)\lim_{x \to c} f(x)limx→c​f(x) exists.

3. lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c)limx→c​f(x)=f(c).

Q3. How can we check the continuity of a function over an interval?

Ans. To check the continuity of a function over an interval, verify that the function is continuous at every point within the interval.

Q4. What is differentiability at a point?

Ans. A function f(x)f(x)f(x) is differentiable at a point x=cx = cx=c if the derivative f′(c)f'(c)f′(c) exists, which means lim⁡h→0f(c+h)−f(c)h\lim_{h \to 0} \frac{f(c+h) - f(c)}{h}limh→0​hf(c+h)−f(c)​ exists.

Q5. Is every continuous function differentiable?

Ans. No, not every continuous function is differentiable. However, every differentiable function is continuous.