# 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)