Question

Show that the function $f\left(x\right)$ defined as
$f\left(x\right)=x\cos \frac{1}{x},x\ne 0$
$=0,x=0$
is continuous at $x=0$ but not differentiable at $x=0$.

Answer 1

It is well know, that

$|\cos a|\le 1$

for every $a\in R$

Because of this it holds

$|x\cos \frac{1}{x}|\le |x|\cdot |\cos \frac{1}{x}|\le |x|\cdot 1=|x|$

Thus

${lim}_{x\to 0}f\left(x\right)=0=f\left(0\right)$

which means that f is continous at $x=0$ but not differentiable.