Projection operator and convex sets

Question

I was wondering if the projection operator onto a convex set was differentiable? [ An explanation would be helpful ] .

Answer 1

The projection operator is not differentiable at the boundary of a set. For example, the projection onto $\mathbb{R}^+$ is $P(x)=\begin{cases}x & \text{if } x>0 \\ 0 & \text{if } x\le 0\end{cases}$, which is not differentiable at $x=0$.