Given a multivariable real-valued function $f$ whose first partials all exist (but which aren't all continuous) at $p$, it is possible that $f$ is not (totally) differentiable at $p$. But since the first partials all exist, grad $f$ is defined at $p$. So we have a non-differentiable function at $p$ that nonetheless has a gradient at $p$. Does this make sense?
The usual definition of stationary point at $p$ is that grad $f(p)=0$, but some texts define stationary point to be $Df(p)=0$. Which is the more fundamental definition? The two definitions are equivalent when $f$ is differentiable, but in the pathological case I painted above where $Df(p)$ doesn't exist but grad $f(p)=0$, does the stationary point for $f$ still exist at $p$? In other words, does stationary point require total differentiability or merely for the gradient to be zero?