Let $U \subset \mathbb{R}^n$ be open and convex, and let $\mathcal{F} \subset C^1(U; \mathbb{R})$ be a family of continuously differentiable functions. If there exists $M > 0$ such that
paragraph
admin
\begin{align*}
|\nabla f(x)| \le M \quad \text{for all } f \in \mathcal{F} \text{ and all } x \in U,
\end{align*}
latex_env
admin
then $\mathcal{F}$ is uniformly equicontinuous on $U$ with modulus $\delta = \varepsilon / M$.