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Question

Let f:RR be a function such that |f(x)|x2, for all xϵR. Then, at x=0, f is :

A
Continuous but not differentiable
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B
Continuous as well as differentiable
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C
Neither continuous nor differentiable
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D
Differentiable but not continuous.
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Solution

The correct option is B Continuous as well as differentiable
We have,
|f(x)|f(x)|f(x)|

Let us consider the continuity of the function at x=0.

We have,

limx0|f(x)|=0 and

limx0|f(x)|=0.

Hence, by applying the sandwich theorem,

limx0f(x)=0

Hence, the function is continuous at x=0.
Now let us consider the differentiability at x=0.

We have,

f(x)=limx0f(x)f(0)x

f(x)=limx0f(x)x

We have,

f(x)xf(x)xf(x)x

As x0, the first and the third term tends to 0 .

Hence,
f(x)=0.
Hence, the function is differentiable at x=0

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