Let \(U\) be an open subset of \(\mathbb{R}^d\), and let \(f : U \to \mathbb{R}\) be a differentiable function. Then, for \(x,y \in U\),
\[ |f(x) - f(y)| \leq \sup_{\xi \in U} \|\nabla f(\xi)\| \, \|x-y\|. \]Let \(U\) be an open subset of \(\mathbb{R}^d\), and let \(f : U \to \mathbb{R}\) be a differentiable function. Then, for \(x,y \in U\),
\[ |f(x) - f(y)| \leq \sup_{\xi \in U} \|\nabla f(\xi)\| \, \|x-y\|. \]