A $C^1$ Map is Differentiable at Every Point

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^{n}$ be an open subset of Euclidean space $\mathbb{R}^{n}$, let $f:U\to\mathbb{R}$ be of class $C^{1}$ on $U$, regarded there as a map into $\mathbb{R}^{m}$ with $m=1$ and single coordinate function $f$, and let $a\in U$.

Then $f$ is differentiable at $a$.