Past exam of the options I course of the University of Oxford / 2012 / F / F1 / a

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,On the open domain, continuous first partial derivatives ensure that this linear expression is the total derivative.