Remark in Shifrin's Multivariable mathematics – math.stackexchange.com

In Ted Shifrin’s multivariable mathematics, the following proposition is given on page 248:

Suppose $U\subseteq \mathbb{R}^n$ is open, $f:U\to \mathbb{R}^m$ is $C^1$ and $a,b\in U$ such that the line ...