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Mathematics Question on Continuity of a function

If f(x)={xe−(1∣x∣+1x);if x≠0 then 0              ;if x=0f(x)=\begin{cases} xe^{-(\frac{1}{|x|}+\frac{1}{x})}; & \text{if } x\ne0 \text{ then} \\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ ; & \text{if }x=0\end{cases}
which of the following is correct ?

A

f(x) is continuous and f'(0) does not exist

B

f(x) is not continuous

C

f(x) is continuous and f'(0) also exists

D

None of these

Answer

f(x) is continuous and f'(0) does not exist

Explanation

Solution

The correct option is (A) : f(x) is continuous and f'(0) does not exist.