= Solution
<Differentiability> here is understood in the sense of the <Fréchet derivative>: there is a <linear map> $L:\mathbb R^n\to\mathbb R^m$ such that
$$
\boxed{f(x+h)=f(x)+Lh+r(h),\qquad \frac{\|r(h)\|}{\|h\|}\longrightarrow0\quad(h\to0).}
$$
The map $L=Df(x)$ is the <Fréchet derivative>. In finite dimensions a <linear map> is bounded, so $\|Lh\|\le C\|h\|$ for some $C$. The defining estimate therefore implies
$$
\|f(x+h)-f(x)\|\le C\|h\|+o(\|h\|)\longrightarrow0.
$$
This is precisely <continuity> at $x$: \b[differentiability implies continuity.]
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