Theorem. Lipschitz condition [br919]

Suppose $\mathbf{f}$ maps a convex open set $E \subset \mathbb{R}^{n}$ into $\mathbb{R}^{m}$, $\mathbf{f}$ is differentiable in $E$, and there is a real number $M$ such that $$ \lVert \mathbf{f}'(\mathbf{x}) \rVert \leq M $$ for every $\mathbf{x} \in E$. Then $$ \lvert \mathbf{f}(\mathbf{b}) - \mathbf{f}(\mathbf{a}) \rvert \leq M \lvert \mathbf{b} - \mathbf{a} \rvert $$ for all $\mathbf{a} \in E, \mathbf{b} \in E$.