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Question

f(x)  = 2x, if x<00, if 0x14x, if x>1


Solution

Here, f(x)  = 2x, if x<00, if 0x14x, if x>1

for x <0,f(x)=2x;0<x<, f(x)=0 and x>1,f(x)=4x are polynomial and constant funtion, so it is a continuous in the given interval. So, we have to check the continuity at x =0, 1.

At x = 0    LHL = limx0 f(x) = limx0 (2x)

Putting x=0-h as x0 when h0 

limx0 [2(0-h)] limx0 (-2h)=2ltimes0=0, 

RHL = limx0+ f(x) = limx0+ (0)=0

Also, f(0)=0   LHL=RHL=f(0)

Thus, f(x) is continuous for all values of x.

At x = 1,    LHL = limx1+ f(x) = limx1+ (0)=0, RHL = limx1+ f(x) = limx1+ (4x)

Putting x=1+h as h1+ when h0

limh0[4(1+h)] = (4+4h)=4+4×0=4

LHLRHL. Thus, f(x) is continuous everywhere except at x=1.

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