Theorem 9.21 Suppose $\mathbf{f}$ maps an open set $E\subset R^n$ into $R^m$. Then $\mathbf{f}\in \mathcal{C}^\prime(E)$ if and only if the partial derivatives $D_jf_i$ exist and are continuous on $E$ for $1\leq i\leq m$, $1\leq j\leq n$.
In the proof, it says "for the converse, it suffices to consider the case $m=1$", but I don't know why. Could you help me? Thanks.
For this direction, we are showing that a differentiable map $f =(f_1, ..., f_m) \in C^1$ if the partial derivatives exist and are continuous. The reason why it suffices to prove this is the product structure of $\mathbb{R}^m$. A bit more specifically, for any differentiable function $E \to \mathbb{R}^m$ and tangent vector $v$ at $p \in E$, we have $D(f)(v) = (D(f_1)(v),...,D(f_m)(v))$.
The map $D(f)$ varies continuously if and only if each $D(f_i)$ vary continuously (this is just saying a matrix varies continuously if and only if each column varies continuously). It thus suffices that each $D(f_j)$ is continuous, and each $f_j$ is a map from $E \to \mathbb{R}$.