Does strict convexity imply differentiability? I know that convexity does not imply differentiability, for example f(x)=|x| is convex but not differentiable. However, |x| is not strictly convex. So I wonder whether strict convexity imply differentiability.
I did some search and found out the Wikipedia implicitly gives the negative answer: http://en.wikipedia.org/wiki/Convex_function#Strongly_convex_functions
It says that "a strongly convex function is also strictly convex" and "a function doesn't have to be differentiable in order to be strongly convex".
Can anyone provide a concrete example? Thanks in advance.

$$f(x)=x^2+|x|$$ is strictly convex because of the $x^2$ term but not differentiable at $0$ because of the $|x|$ term
