$|x|^{|x|}$ is continuous in $\mathbb{R}$

Question

I'm trying to show this now my self, but still no go. There isn't really a concrete attempt that I can show..

Help?

Answer 1

Hint $|x|^{|x|}=\exp(f(x))$ where $$ f(x)=\begin{cases} |x|\log |x| &\text{if }x\neq 0\\ 0 & \text{if }x=0 \end{cases} $$

Show that $f$ is continuous.