Is a convex function always twice differentiable?
Solution 1:
What about $f(x)=\|x\|$? (Euclidean norm)
Solution 2:
Convexity doesn't even imply continuity(e.g one can construct a convex function on a closed domain which is nowhere continuous on the boundary).
Edit
However, Alexandrov's Theorem states that a convex function is $\mathcal C^2$ almost-everywhere. Now, that's alot of differentiability (depending on what you want to do with it)!