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

In the previous class, you have already learned about limits, derivatives, limits, and derivatives of trigonometric functions. In this article, you will get NCERT solutions for Class 12 Maths chapter 5 exercise 5.1. NCERT book Class 12 Maths chapter 5 exercise 5.1 consists of questions related to finding whether a function is continuous or not.

Continuity of functions can't be learned without fundamental knowledge of limit which you have learned already. It is a fundamental concept of calculus that you must know to understand more concepts of calculus. Solving exercise 5.1 Class 12 Maths questions are very important to get conceptual clarity about continuity. There are different theorems to check the continuity of different types of functions mentioned in the NCERT syllabus of Class 12 Maths. You must go through these theorems also. Check here for NCERT Solutions.

**Also, see**

- Continuity and Differentiability Exercise 5.2
- Continuity and Differentiability Exercise 5.3
- Continuity and Differentiability Exercise 5.4
- Continuity and Differentiability Exercise 5.5
- Continuity and Differentiability Exercise 5.6
- Continuity and Differentiability Exercise 5.7
- Continuity and Differentiability Exercise 5.8
- Continuity and Differentiability Miscellaneous Exercise

**Continuity and Differentiability Exercise: 5.1**

**Question:1**. Prove that the function is continuous at and at

**Answer:**

Given function is

Hence, function is continous at x = 0

Hence, function is continous at x = -3

Hence, function is continuous at x = 5

**Question:2**.Examine the continuity of the function

**Answer:**

Given function is

at x = 3

Hence, function is continous at x = 3

**Question:3 **Examine the following functions for continuity.

**Answer:**

Given function is

Our function is defined for every real number say k

and value at x = k ,

and also,

Hence, the function is continuous at every real number

**Question:3 b)** Examine the following functions for continuity.

**Answer:**

Given function is

For every real number k ,

We get,

Hence, function continuous for every real value of x,

**Question:3 c)** Examine the following functions for continuity.

**Answer:**

Given function is

For every real number k ,

We gwt,

Hence, function continuous for every real value of x ,

**Question:3 d)** Examine the following functions for continuity.

**Answer:**

Given function is

for x > 5 , f(x) = x - 5

for x < 5 , f(x) = 5 - x

SO, different cases are their

case(i) x > 5

for every real number k > 5 , f(x) = x - 5 is defined

Hence, function f(x) = x - 5 is continous for x > 5

case (ii) x < 5

for every real number k < 5 , f(x) = 5 - x is defined

Hence, function f(x) = 5 - x is continous for x < 5

case(iii) x = 5

for x = 5 , f(x) = x - 5 is defined

Hence, function f(x) = x - 5 is continous for x = 5

Hence, the function is continuous for each and every real number

**Question:4**. Prove that the function is continuous at x = n, where n is a positive integer

**Answer:**

GIven function is

the function is defined for all positive integer, n

Hence, the function is continuous at x = 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?

**Answer:**

Given function is

function is defined at x = 0 and its value is 0

Hence , given function** is continous at **x = 0

given function is defined for x = 1

Now, for x = 1 Right-hand limit and left-hand limit are not equal

R.H.L L.H.L.

Therefore, given function is **not continous **at x =1

Given function is defined for x = 2 and its value at x = 2 is 5

Hence, given function** is continous** at x = 2

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

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 2

Hence, given function is continuous for each value of k > 2

case(ii) k < 2

Hence, given function is continuous for each value of k < 2

case(iii) x = 2

Right hand limit at x= 2 Left hand limit at x = 2

Therefore, x = 2 is the point of discontinuity

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

**Answer:**

Given function is

GIven function is defined for every real number k

Different cases are their

case (i) k < -3

Hence, given function is continuous for every value of k < -3

case(ii) k = -3

Hence, given function is continous for x = -3

case(iii) -3 < k < 3

Hence, for every value of k in -3 < k < 3 given function is continous

case(iv) k = 3

Hence**. x = 3 is the point of discontinuity**

case(v) k > 3

Hence, given function is continuous for each and every value of k > 3

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

**Answer:**

Given function is

if x > 0 ,

if x < 0 ,

given function is defined for every real number k

Now,

case(i) k < 0

Hence, given function is continuous for every value of k < 0

case(ii) k > 0

Hence, given function is continuous for every value of k > 0

case(iii) x = 0

Hence,** 0 is the only point of discontinuity**

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

**Answer:**

Given function is

if x < 0 ,

Now, for any value of x, the value of our function is -1

Therefore, the given function is continuous for each and every value of x

Hence, no point of discontinuity

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

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 1

Hence, given function is continuous for each value of k > 1

case(ii) k < 1

Hence, given function is continuous for each value of k < 1

case(iii) x = 1

Hence, at x = 2 given function is continuous

Therefore, no point of discontinuity

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

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 2

Hence, given function is continuous for each value of k > 2

case(ii) k < 2

Hence, given function is continuous for each value of k < 2

case(iii) x = 2

Hence, given function is continuous at x = 2

There, no point of discontinuity

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

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 1

Hence, given function is continuous for each value of k > 1

case(ii) k < 1

Hence, given function is continuous for each value of k < 1

case(iii) x = 1

Hence, x = 1 is the point of discontinuity

**Question:13.** Is the function defined by

a continuous function?

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 1

Hence, given function is continuous for each value of k > 1

case(ii) k < 1

Hence, given function is continuous for each value of k < 1

case(iii) x = 1

Hence, x = 1 is the point of discontinuity

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

**Answer:**

Given function is

GIven function is defined for every real number k

Different cases are their

case (i) k < 1

Hence, given function is continous for every value of k < 1

case(ii) k = 1

Hence, given function is discontinous at x = 1

Therefore, x = 1 is he point od discontinuity

case(iii) 1 < k < 3

Hence, for every value of k in 1 < k < 3 given function is continous

case(iv) k = 3

Hence. x = 3 is the point of discontinuity

case(v) k > 3

Hence, given function is continous for each and every value of k > 3

case(vi) when k < 3

Hence, for every value of k in k < 3 given function is continous

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

**Answer:**

Given function is

Given function is satisfies for the all real values of x

case (i) k < 0

Hence, function is continuous for all values of x < 0

case (ii) x = 0

L.H.L at x= 0

R.H.L. at x = 0

L.H.L. = R.H.L. = f(0)

Hence, function is continuous at x = 0

case (iii) k > 0

Hence , function is continuous for all values of x > 0

case (iv) k < 1

Hence , function is continuous for all values of x < 1

case (v) k > 1

Hence , function is continuous for all values of x > 1

case (vi) x = 1

Hence, function is not continuous at x = 1

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

**Answer:**

Given function is

GIven function is defined for every real number k

Different cases are their

case (i) k < -1

Hence, given function is continuous for every value of k < -1

case(ii) k = -1

Hence, given function is continous at x = -1

case(iii) k > -1

Hence, given function is continous for all values of x > -1

case(vi) -1 < k < 1

Hence, for every value of k in -1 < k < 1 given function is continous

case(v) k = 1

Hence.at x =1 function is continous

case(vi) k > 1

Hence, given function is continous for each and every value of k > 1

case(vii) when k < 1

Hence, for every value of k in k < 1 given function is continuous

Therefore, continuous at all points

**Question:17.** Find the relationship between a and b so that the function f defined by

is continuous at x = 3.

**Answer:**

Given function is

For the function to be continuous at x = 3 , R.H.L. must be equal to L.H.L.

For the function to be continuous

**Question:18.** For what value of l is the function defined by

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

**Answer:**

Given function is

For the function to be continuous at x = 0 , R.H.L. must be equal to L.H.L.

For the function to be continuous

Hence, for no value of function is continuous at x = 0

For x = 1

Hence, given function is continuous at x =1

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

**Answer:**

Given function is

Given is defined for all real numbers k

Hence, by this, we can say that the function defined by is discontinuous at all integral points

**Question:20.** Is the function defined by continuous at x = ?

**Answer:**

Given function is

Clearly, Given function is defined at x =

Hence, the function defined by continuous at x =

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

a)

**Answer:**

Given function is

Given function is defined for all real number

We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

We proved independently that sin x and cos x is continous function

So, we can say that

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

**Question:21. b)**** **Discuss the continuity of the following functions:

**Answer:**

Given function is

Given function is defined for all real number

We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

We proved independently that sin x and cos x is continous function

So, we can say that

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

**Question:21 c)** Discuss the continuity of the following functions:

**Answer:**

Given function is

Given function is defined for all real number

We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

We proved independently that sin x and cos x is continous function

So, we can say that

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

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

**Answer:**

We, know that if two function g(x) and h(x) are continuous then

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, the function is a continuous function

We proved independently that sin x and cos x is a continous function

So, we can say that

cosec x = is also continuous except at

sec x = is also continuous except at

cot x = is also continuous except at

**Question:23.** Find all points of discontinuity of f, where

**Answer:**

Given function is

Hence, the function is continuous

Therefore, no point of discontinuity

**Question:24. **Determine if f defined by

is a continuous function?

**Answer:**

Given function is

Given function is defined for all real numbers k

when x = 0

Hence, function is continuous at x = 0

when

Hence, the given function is continuous for all points

**Question:25**. Examine the continuity of f, where f is defined by

**Answer:**

Given function is

Given function is defined for all real number

We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

We proved independently that sin x and cos x is continous function

So, we can say that

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

When x = 0

Hence, function is also continuous at x = 0

**Question:26.** Find the values of k so that the function f is continuous at the indicated point in Exercises

**Answer:**

Given function is

When

For the function to be continuous

Therefore, the values of k so that the function f is continuous is 6

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

**Answer:**

Given function is

When x = 2

For the function to be continuous

f(2) = R.H.L. = LH.L.

Hence, the values of k so that the function f is continuous at x= 2 is

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

**Answer:**

Given function is

When x =

For the function to be continuous

f() = R.H.L. = LH.L.

Hence, the values of k so that the function f is continuous at x= is

**Question:29** Find the values of k so that the function f is continuous at the indicated point in Exercises

**Answer:**

Given function is

When x = 5

For the function to be continuous

f(5) = R.H.L. = LH.L.

Hence, the values of k so that the function f is continuous at x= 5 is

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

is a continuous function.

**Answer:**

Given continuous function is

The function is continuous so

By solving equation (i) and (ii)

a = 2 and b = 1

Hence, values of a and b such that the function defined by is a continuous function is 2 and 1 respectively

**Question:31.** Show that the function defined by is a continuous function.

**Answer:**

Given function is

given function is defined for all real values of x

Let x = k + h

if

Hence, the function is a continuous function

**Question:32.** Show that the function defined by is a continuous function.

**Answer:**

Given function is

given function is defined for all values of x

f = g o h , g(x) = |x| and h(x) = cos x

Now,

g(x) is defined for all real numbers k

case(i) k < 0

Hence, g(x) is continuous when k < 0

case (ii) k > 0

Hence, g(x) is continuous when k > 0

case (iii) k = 0

Hence, g(x) is continuous when k = 0

Therefore, g(x) = |x| is continuous for all real values of x

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

g(x) is continuous , h(x) is continuous

Therefore, f(x) = g o h is also continuous

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

**Answer:**

Given function is

f(x) = sin |x|

f(x) = h o g , h(x) = sin x and g(x) = |x|

Now,

g(x) is defined for all real numbers k

case(i) k < 0

Hence, g(x) is continuous when k < 0

case (ii) k > 0

Hence, g(x) is continuous when k > 0

case (iii) k = 0

Hence, g(x) is continuous when k = 0

Therefore, g(x) = |x| is continuous for all real values of x

Now,

h(x) = sin x

Let suppose x = c + h

if

Hence, function is a continuous function

g(x) is continuous , h(x) is continuous

Therefore, f(x) = h o g is also continuous

**Question:34. **Find all the points of discontinuity of f defined by

**Answer:**

Given function is

Let g(x) = |x| and h(x) = |x+1|

Now,

g(x) is defined for all real numbers k

case(i) k < 0

Hence, g(x) is continuous when k < 0

case (ii) k > 0

Hence, g(x) is continuous when k > 0

case (iii) k = 0

Hence, g(x) is continuous when k = 0

Therefore, g(x) = |x| is continuous for all real values of x

Now,

g(x) is defined for all real numbers k

case(i) k < -1

Hence, h(x) is continuous when k < -1

case (ii) k > -1

Hence, h(x) is continuous when k > -1

case (iii) k = -1

Hence, h(x) is continuous when k = -1

Therefore, h(x) = |x+1| is continuous for all real values of x

g(x) is continuous and h(x) is continuous

Therefore, f(x) = g(x) - h(x) = |x| - |x+1| is also continuous

**More About NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1:-**

In Class 12th Maths chapter 5 exercise 5.1 there are 34 long answer types questions only checking your knowledge of continuity. Many questions in Class 12 Maths ch 5 ex 5.1 are related to checking the continuity of trigonometric functions. There are 20 examples and some important theorems given before this exercise in the NCERT textbook. Solving these examples is a must to do before going to the Class 12th Maths chapter 5 exercise 5.1 questions because it will help you in solving NCERT problems.

**Also Read|** Continuity and Differentiability Class 12th Chapter 5 Notes

**Benefits of ****NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1:-**

**Benefits of**

**NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1:-**

- Class 12 Maths chapter 5 exercise 5.1 solutions are described in a detailed manner, so you will get the concept of continuity easily.
- In Class 12 Maths chapter 5 exercise 5.1 solutions, you will get different ways to approach the problem.
- There are some properties of continuous functions which make it easy to check the continuity of the given functions.
- In Class 12 Maths chapter 5 exercise 5.1 solutions, you will learn about the algebra of continuous functions.

**Also see-**

NCERT Solutions for Class 12 Maths Chapter 5

NCERT Exemplar Solutions Class 12 Maths Chapter 5

**NCERT Solutions of Class 12 Subject Wise**

NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12 Physics

NCERT Solutions for Class 12 Chemistry

NCERT Solutions for Class 12 Biology

**Subject Wise NCERT Exampler Solutions**

NCERT Exemplar Solutions for Class 12th Maths

NCERT Exemplar Solutions for Class 12th Physics

NCERT Exemplar Solutions for Class 12th Chemistry

NCERT Exemplar Solutions for Class 12th Biology

**Happy learning!!!**

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