# Exercise 9.2

Question 1

Prove that the function is everywhere continuous .

Sol :

When , we have

We know that sin x as well as the identity function are everywhere continuous .

So , the quotient function is continuous at each point

When , we have , which is a polynomial function .

Therefore, is continuous at each point

Now , Let us consider the point

Given ,

We have , (LHL at x = 0)

= 1

(RHL at x = 0)

= 1

Also ,

Thus , is continuous at 0

Hence, is everywhere continuous

Question 2

Discuss the continuity of the function

Sol :

We have (LHL at x = 0)

= -1

(RHL at x = 0)

= 1

Thus , is discontinuous at

Question 3

Find the points of discontinuity, if any, of the following functions:

(i)

Sol :

When then

We know that a polynomial function is everywhere continuous

So , is continuous at each point at

Now , at we have

(LHL at x = 1)

=0

( RHL at x = 1 )

=0

Also ,

Thus , is discontinuous at

Hence , the only point of discontinuity for

(ii) (ii)

Sol :

When , then

We know that a polynomial function is everywhere continuous . Therefore , the functions are everywhere continuous .

So , the product function is continuous at every

Now at x = 2 , we have

(LHL at x=2)

= 32

(RHL at x=2)

= 32

Also ,

Thus , is discontinuous for

(iii)