Past exam of the options II for mathematics and philosophy course of the University of Oxford / 2013 / F2 / a / ii / Solution

The map is continuously differentiable if it is differentiable at every point of $U$ and the derivative map $x\mapsto D{F}_x$ is continuous. In finite dimensions this is equivalent to all first partial derivatives of all its components existing and being continuous on $U$. The class is denoted $C^1(U,{\mathbb{R}}^m)$.