This is a follow-up question to:
Derivative Of A Function Defined In Terms Of An Infimum
Let $A$ be a compact, (smooth) embedded submanifold of $\mathbb{R}^n$ and let $\Phi : \mathbb{R}^n \rightarrow \mathbb{R}$ be defined by $\displaystyle \Phi(x) = \inf_{p \in A} ||p - x||_2^2$. Does $\Phi$ have a total derivative everywhere? How does one go about finding the total derivative of $\Phi$? Is it possible to interchange the infimum and a partial derivative?