Let $X \subset \mathbb{R}^{n}$ be compact and convex. Moreover, let $f:X \rightarrow \mathbb{R}$ be a differentiable map with $\sup\_{x \in X} \|\nabla f(x)\| = K < \infty$, where $\|\cdot\|$ denotes the Euclidean norm and $\nabla f$ the gradient. 


Both [in this question](https://mathoverflow.net/questions/115458/smallest-lipschitz-constant-on-non-convex-domains) and [in this webpage](https://www.encyclopediaofmath.org/index.php/Lipschitz_constant) it is said that the best (minimal) Lipschitz constant of $f$ is $K$. If this is a basic result in real analysis, could someone please point me to a reference?



Smallest Lipschitz Constant of a Differentiable Function
This is essentially Corollary 1.42 in my book *Lipschitz Algebras* (second edition).


