Solution
= Solution
The function is <continuously differentiable> when all its first <partial derivatives> exist on $U$ and are continuous there. Equivalently, $h$ has a <Fréchet derivative> at every point and the map $x\mapsto Dh(x)$ is continuous. For a scalar-valued function,
$$
Dh(x)v=\sum_{j=1}^n\partial_jh(x)v_j.
$$
On the open domain, continuous first <partial derivatives> ensure that this linear expression is the total <derivative>.