Let $f \in C^1([0,T]\times S(s))$ where $S(r) \subset \mathbb{R}^n$ is compact for each $r \in [0,T].$
Does it not automatically follow that $f \in C([0,T], C^1(S(s)))$?
In the paper I'm reading, apparently it is true only because $f$ and its first derivative wrt. $x$ are continuous and thus uniformly continuous on the compact set $[0,T] \times S(s)$. Can someone explain this to me please?