Let $x_1,x_2,...,x_n$ be n points in $\mathbb{R}^m$. Is the function $F(w)=max(w^Tx_1,w^Tx_2,...,w^Tx_n)$ differentiable for all $w$?

Question

We can assume $x_i \neq x_j$ if $i \neq j$. My hunch is this function is differentiable everywhere except at $w=0$ as it is just maximum over a bunch of linear functions.

Answer 1

Consider $m=2$, and $x_1 = (1,0)$, $x_2 = (-1,0)$.

Then, your function will be $F(w)= F((w_x,w_y)) = \max(x_1 \cdot w, x_2 \cdot w_2) = \max(w_x,-w_x) = |w_x|$. This will not be differentiable along the whole crease $w_x =0$, even for non-zero $w_y$.