Why is $x\ln|x|$ not differentiable at 0? My professor asked me to think of a function that is continuous from $\mathbb{R} \rightarrow \mathbb{R}$ whose derivative is not continuous. I thought of the function $f(x)=x\ln|x|, f(0)=0$ but was told it wasn't differentiable at 0. Could anyone show me why?

Hint. As $x \to 0^+$,
$$
\frac{f(x)-0}{x-0}=\frac{x\ln x-0}{x}=\ln x \to -\infty
$$ which is not finite.
