For each $t$, let $u(t):\Omega \to \mathbb{R}$ be a function defined on a bounded domain $\Omega \subset \mathbb{R}^n$. We can think of this as $$u(t)(x) = u(t,x)$$ for $x \in \Omega.$
If $\nabla u \in L^\infty(0,T;L^\infty(\Omega))$ (this is the spatial gradient) can I say anything about $u$? I'd like to say $u$ is continuous in time or something like that?