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Question

f(x) = {x+1, if x 1x2+1, if x<1


Solution

Here,    f(x) = {x+1, if x 1x2+1, if x<1

for x > 1, f(x) = x + 1 and x < 1, f(x) = x2 + 1 is a polynomial funtion, so f(x) is a continuous in the given interval. Therefore, we have to check the continuity at x = 1.

LHL = limx1 f(x) = limx1 x2+1

Putting x=-1+h as x1 when h0

limh0[(1h)2+1]limh0[1+h22h+1]limh0[2+h22h]= 2+0-0=2

RHL = limx1+ f(x) = limx1+ (x+1)

Putting x=1+h as x1+ when x0

limh0 (1+h+1) = limh0 (2+h)=2+0=2

Also   f(1)=1+1=2    [f(x)=x+1]

LHL = RHL = f(1). Thus, f(x) is continuous at x=1.

Hence, there is no point of discontinuity for this function f(x).

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