Suppose f is a continuous function on R such that fx Solutio

Suppose f is a continuous function on R such that |f(x)|

Solution

(a)

The given function f is

x<0 or IxI= -x and x>0 or IxI= x

therefore,

at x=0, IxI=0 or f(0)=I0I=0

(b) It is given that a<b

and If(x)I<=LIxI

f(a)=IaI and f(b)= IbI

or f(0)=0 and f(1)=I1I=1

It is possible only when there is another function L belongs to (0,1)

Or it is proved.

0<1

we can say a=0 and b=1