Mean value inequality

Mean value inequality: Let $U\subseteq\mathbb R^k$ be an open set , let $f:U\to\mathbb R^m$ be continuously differentiable, and let $x,y\in U$ be such that the line segment $\{(1-t)x+ty:0\le t\le 1\}$ is contained in $U$. If there is $M\ge 0$ with

where $\|Df(z)\|$ is the operator norm of the derivative, then

For $k=m=1$ this recovers the familiar estimate $|f(y)-f(x)|\le \sup |f'|\cdot |y-x|$ derived from the mean value theorem .