# Class 12 ncert solution math exercise 5.1

# Exercise 5.1 (Continuity and Differentiability)

**Question 1: Prove that the function** **is continuous at** **and at** . (Class 12 ncert solution math exercise 5.1)

**Solution: The given function is**

At

Therefore, is continous at .

Therefore, is continous at .

Therefore, is continous at .

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

**Solution:**The given function is

At

Therefore, is continous at .

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

(i)

(ii)

(iii)

(iv)

**Solution:(i) The given function is**

It is evident that is defined at every real number and its value at is .

It is also observed that

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

**(ii) The given function is** For any real number , we obtain

Also,

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

**(iii) The given function is**

For any real number , we obtain

Also,

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

**(iv) The given function is**

This function is defined at all points of the real line. Let be a point on a real line. Then, or

**Case I:**

Then,

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

**Case II:**

Then,

Therefore, is continuous at

**Case III:**

Then,

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

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

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

**Solution: The given function is**

It is observed that is defined at all positive integers, , and its value at is .

Then,

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

**Question 5: Is the function defined by** **continuous at** At At

**Solution: The given function is**

At ,

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

Then,

Therefore, is continuous at .

At ,

It is evident that is defined at 1 and its value at 1 is 1 .

The left hand limit of at is,

The right hand limit of at is,

Therefore, is not continuous at .

At ,

It is evident that is defined at 2 and its value at 2 is 5 .

Therefore, is continuous at .

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

**Solution: The given function is**

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

Let be a point on the real line. Then three case arise

**Case I:**

**Case II:**

**Case III:**

**Case I:**

Then,

Therefore, is continuous at all points , such that .

**Case II:**

Then,

Therefore, is continuous at all points , such that

**Case III:**

Then, the left hand limit of at is,

The right hand limit of at is,

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

Therefore, is not continuous at .

Hence, is the only point of discontinuity of .

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

**Solution:**The given function is

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

Let be a point on the real line.

**Case I:**If , then

Therefore, is continuous at all points , such that .

**Case II:**

If , then

Therefore, is continuous at .

**Case III:**If , then

Therefore, is continuous in .

**Case IV:**If , then the left hand limit of at is,

The right hand limit of at is,

It is observed that the left and right hand limit of at do not coincide. Therefore, is not continuous at

.

**Case V:**If , then

Therefore, is continuous at all points , such that .

Hence, is the only point of discontinuity of .

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

**Solution: The given function is**

It is known that, and

Therefore, the given function can be rewritten a

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

Let be a point on the real line.

**Case I:**If , then

Therefore, is continuous at all points .

**Case II:**If , then the left hand limit of at is,

The right hand limit of at is,

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

Therefore, is not continuous at .

**Case III:**If , then

Therefore, is continuous at all points , such that .

Hence, is the only point of discontinuity of .

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

**Solution: **

The given function is

It is known that

Therefore, the given function can be rewritten as

Let be any real number.

Then,

Also,

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 , where is defined by

**Solution:** The given function is

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

Let be a point on the real line.

Case I: If , then

Therefore, is continuous at all points , such that .

**Case II: **If , then

The left hand limit of at is,

The right hand limit of at is,

Therefore, is continuous at .

**Case III:** If , then

Therefore, is continuous at all points , such that .

Hence, the given function has no point of discontinuity.

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

Solution: The given function is

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

Let be a point on the real line.

**Case I: **If , then

Therefore, is continuous at all points , such that .

**Case II:** If , then

Therefore, is continuous at .

**Case III:** If , then

Therefore, is continuous at all points , such that .

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

Hence, has no point of discontinuity.

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

**Solution: **The given function is

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

Let be a point on the real line.

**Case I:**If , then

Therefore, is continuous at all points , such that .

**Case II:** If , then the left hand limit of at is,

The right hand limit of at is,

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

Therefore, is not continuous at .

**Case III:** If , then

Therefore, is continuous at all points , such that .

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

**Question 13:** Is the function defined by a continous function?

**Solution:** The given function is

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

Let be a point on the real line.

**Case I:** If , then

Therefore, is continuous at all points , such that .

**Case II:** If , then

The left hand limit of at is,

The right hand limit of at is,

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

Therefore, is not continuous at .

**Case III: **If , then

Therefore, is continuous at all points , such that .

From the above observation it can be concluded that, is the only point of discontinuity of .

**Question 14:**

Discuss the continuity of the function , where is defined by

**Solution:** The given function is

The given function is defined at all the points of the interval .

Let be a point in the interval .

Case I: If , then

Therefore, is continuous in the interval .

**Case II: **If , then

The left hand limit of at is,

The right hand limit of at is,

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

Therefore, is not continuous at .

**Case III:** If , then

Therefore, is continuous at in the interval .

**Case IV:** If , then

The left hand limit of at is,

The right hand limit of at is,

It is observed that the left and right hand limit of at do not coincide. Therefore, is discontinuous at .

**Case V:**If , then

Therefore, is continuous at all points of the interval .

Hence, is discontinuous at and .

**Question 15: **

**Solution:Â ** The given function is

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

**Case I:** If , then

The left hand limit of at is,

The right hand limit of at is,

Therefore, is continuous at

**Case II:** If , then

The left hand limit of at is,

The right hand limit of at is,

It is observed that the left and right hand limit of at

do not coincide.

Therefore, is not continuous at .

Therefore, is continuous at all points , except .

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

**Solution:** The given function is

The given function is defined at all the points.

**Case I: **If , then

herefore, is continuous at all points , such that .

**Case II:** If , then

The left hand limit of at is,

The right hand limit of at is,

Therefore, is continuous at

**Case III:** If , then

The left hand limit of at is,

The right hand limit of at is,

Therefore, is continuous at .

Therefore, is continuous at all points , such that .

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

**Question 17:** **Find the relationship between** **and** **so that the function** ** defined by** is continous at .

**Solution:** The given function is

For to be continuous at , then

Also,

Therefore, from (1), we obtain

Therefore, the required relationship is given by, .

**Question 18: **For what value of is the function defined by is continous at What about continuity at

**Solution:** **The given function is**

If is continuous at , then

Therefore, there is no value of for which is

continuous at .

At x=1

f(1)=4 x+1=4(1)+1=5

Therefore, for any values of is continuous at .

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

**Solution:** The given function is

It is evident that is defined at all integral points.

Let be an integer.cuemath

Then,

The left hand limit of at is,

The right hand limit of at is,

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

Therefore, is not continuous at .

Hence, is discontinuous at all integral points.

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

**Solution:** The given function is

It is evident that is defined at .

At

Consider

Put , it is evident that if , then

Therefore, the given function is continuous at .

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

**(i)**

**(ii)**

**(iii)**

**Solution:** It is known that if and are two continuous functions, then and are also continuous.

Let and are continuous functions.

It is evident that is defined for every real number.

Let be a real number. Put

If , then

Therefore, is a continuous function.

Let

It is evident that is defined for every real number.

Let be a real number. Put

If , then

Therefore, is a continuous function.

Therefore, it can be concluded that,

**(i)** is a continuous function.

**(ii)** is a continuous function.

**(iii)** is a

continuous function.

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

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

is continuous.

is continuous.

is continuous.

Let and are continuous functions.

It is evident that is defined for every real number.

Let be a real number. Put

If , then

Therefore, is a continuous function.

Let

It is evident that is defined for every real number.

Let be a real number. Put

If , then

Therefore, is a continuous function.

Therefore, it can be concluded that, is continuous.

is continuous.

Therefore, cosecant is continuous except at

is continuous.

is continuous.

Therefore, secant is continuous except at is continuous. is continuous.

Therefore, cotangent is continuous except at .

**Question 23:**

**Find the points of discontinuity of** , where

**Solution:** The given function is

The given function is defined at all the points of the real line. Let be a point on the real line.

**Case I:** If , then

Therefore, is continuous at all points , such that .

**Case II:** If , then

Therefore, is continuous at all points , such that .

**Case III:** If , then

The left hand limit of at is,

The right hand limit of at is,

Therefore, is continuous at

From the above observations, it can be concluded that is

continuous at all points of the real line.

Thus, has no point of discontinuity.

**Question 24:** Determine if defined by , if is a continuous function?

**Solution:**

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

Let be a point on the real line.

**Case I:** If , then

Therefore, is continuous at all points , such that .

**Case II:** If , then

It is known that,

Similarly,

Therefore, is continuous at .

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

Thus, is a continuous function.

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

**Solution:** The given function is

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

Let be a point on the real line.

Case I:If , then

Therefore, is continuous at all points , such that .

Case II:If , then

Therefore, is continuous at .

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

Thus, is a continuous function.

**Question 26:** **Find the values of** **so that the function is continuous at the indicated point** at

**Solution:** The given function is

The given function is continuous at , if is defined at and if the value of the at equals the limit of at .

It is evident that is defined at and

Put

Then

Therefore, the value of .

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

**Solution:** The given function is

The given function is continuous at , if is defined at and if the value of the at equals the limit of at .

It is evident that is defined at and

Therefore, the value of .

**Question 28: Find the values of** ** so that the function is continuous at the indicated point** at

**Solution:** The given function is

The given function is continuous at , if is defined at and if the value of the at equals the limit of at .

It is evident that is defined at and

.

Therefore, the value of .

**Question 29: Find the values of** **so that the function** **is continuous at the indicated point** at .

**Solution:** The given function is

The given function is continuous at , if is defined at and if the value of the at equals the limit of at .

It is evident that is defined at and

Therefore, the value of .

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

**is a continuous function.**

**Solution:**

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

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

In particular, is continuous at and

Since is continuous at , we obtain

Since is continuous at , we obtain

On subtracting equation (1) from equation , we obtain

By putting in equation (1), we obtain

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

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

**Solution:** The given function is .

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

, where and

It has to be proved first that and are continuous functions.

It is evident that is defined for every real number.

Let be a real number.

Let . Put

If , then

Therefore, is a continuous function.

Let

It is evident that is defined for every real number.

Let be a real number, then

Therefore, is a continuous function.

It is known that for real valued functions and , such that is defined at , if is continuous at and if is continuous at , then is continuous at .

Therefore, is a continuous function.

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

**Solution:** The given function is .

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

, where and

It has to be proved first that and are continuous functions.

It is evident that is defined for every real number.

Let be a real number.

**Case I:** If , then

Therefore, is continuous at all points , such that .

**Case II:**If , then

Therefore, is continuous at all points , such that .

**Case III:** If , then

.

Therefore, is continuous at all .

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

It is evident that is defined for every real number.

Let be a real number. Put

If , then

Therefore, is a continuous function.

It is known that for real valued functions and , such that is defined at , if