Are there any functions F with no gradient at some point?
@theyComeAndGo Sure: the $L^1$ norm defines a function
$$F(f) := \int_\Omega |f(p)|\ dp,$$
which looks in some sense like the ordinary absolute value function $|x|$ on the real line; in particular, it's not differentiable at the origin, since no matter which direction you go, the function increases. This is also true for the $L^2$ norm, but in this case $F$ is "flat" at the origin, i.e., an infinitesimal motion in any direction doesn't increase the norm at all (i.e., there is no 1st-order change in the function).