Mean value inequality

Mean value inequality. Let $U\subseteq\mathbb R^k$ be open, let $f:U\to\mathbb R^m$ be continuously differentiable, and suppose the line segment $[x,y]$ lies in $U$ . If $M\ge0$ satisfies

\|Df(z)\|\le M \quad \text{for all } z \text{ on the segment from } x \text{ to } y,

where $\lVert Df(z)\rVert$ is the operator norm, then

\|f(y)-f(x)\|\le M\|y-x\|.

Remarks

For $k=m=1$ , this gives $|f(y)-f(x)|\le M|y-x|$ .