 Solution

The map $f:U\to {\mathbb{R}}^n$ is differentiable at $a$ when there is a linear map $Df(a)$ such thatIt is $C^1$ when it is differentiable everywhere and $a\mapsto Df(a)$ is continuous. The definition immediately gives $\Vert f(a+h)-f(a)\Vert \le (\Vert Df(a)\Vert +o(1))\Vert h\Vert \to 0$, so differentiability implies continuity.

The inverse function theorem says that if $f$ is $C^1$ near $a$ and $Df(a)$ is invertible, then $f$ restricts to a diffeomorphism between neighbourhoods of $a$ and $f(a)$.