Understanding theorem in Rudin’s Principles of Mathematical Analysis Theorem 9.21

Question

Suppose $\textbf{f}$ maps an open set $E \subset \mathbb{R}^n $ into $\mathbb{R}^m$. Then $\textbf{f} \in C'(E)$ if and only if the partial derivatives $D_jf_i$ exist and are continuous on $E$ for $1\leq i \leq m ,1\leq i \leq n $.

This is theorem 9.21 of Rudin 3rd edition (page 219).

In the proof, for the converse part rudin proved the result only for $m=1$. How we can prove the converse part for $m>1$

Answer 1

Let $\textbf{f}=(f_1,...,f_m):E \to \mathbb R^m$ with $f_j: E \to \mathbb R$.

Then we have;

$\textbf{f} \in C^1(E, \mathbb R^m) \iff f_1,...,f_m \in C^1(E, \mathbb R).$