Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}^m$ be a map. Then $f \in C^1(U;\mathbb{R}^m)$ if and only if $f$ is differentiable at every point $a \in U$ and the derivative map
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\begin{align*}
Df: U &\to \mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)
\end{align*}
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given by $a \mapsto Df_a$ is continuous with respect to the operator norm topology on $\mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$.