Let f,g := $\displaystyle R \rightarrow R$ be given by

f(x) :=

$\displaystyle x sin(\frac{1}{x}), x \neq 0$

0, x = 0

and

g(x) :=

$\displaystyle x^2 sin(\frac{1}{x}), x \neq 0$

0, x = 0

Prove that f is not differentiable at x = 0 and that g is differentiable at x = 0.