If f(x)=xsin⁡(1/x), for x≠00, for x=0 then

If $f (x) = \{\begin{cases} x \sin (1 / x), for x \neq 0 \\ 0, for x = 0 \end{cases}$ then

A
$f$ f is a continuous function
B
$f^{'} (0 +)$ exists but $f^{'} (0 -)$ does not exist
C
$f^{'} (0 +) = f^{'} (0 -)$
D
$f^{'} (0 -)$ exists but $f^{'} (0 +)$ does not exist.

Solution:

$f$ is continuous $f^{'} (0 +) = lim_{x \to 0 +} \sin \frac{1}{x}$ which does not exist. Similarly $f^{'} (0 -)$ does not exist.

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