= Solution
The map $f:U\to\mathbb R^n$ is <differentiable function>[differentiable] at $a$ when there is a linear map $Df(a)$ such that
$$
f(a+h)=f(a)+Df(a)h+o(\|h\|).
$$
It is $C^1$ when it is differentiable everywhere and $a\mapsto Df(a)$ is continuous. The definition immediately gives $\|f(a+h)-f(a)\|\leq(\|Df(a)\|+o(1))\|h\|\to0$, 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 $f(a)$.
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