TITLE: Is $f(x)=x|x|$ differentiable everywhere?
QUESTION [7 upvotes]: When $f$ is a function $\mathbf{R}$ to $R$. I know $\lim_{x \to 0+}\frac{f(x)-f(0)}{x-0}=
\lim_{x \to 0+}\frac{x^2}{x}=0$ and $\lim_{x \to 0-}\frac{f(x)-f(0)}{x-0}=
\lim_{x \to 0-}\frac{-x^2}{x}=0$, but $|x|$ is not differentiable everywhere, so I'm doubting myself.

REPLY [4 votes]: It should be clear that $f$ is differentiable for $x\neq 0$.
For $x=0$, we have
$\lim_{x \to 0} { f(x)-f(0) \over x-0} = \lim_{x \to 0} |x| = 0$, hence $f$
is differentiable at $x=0$ with derivative zero.