# Smallest Lipschitz Constant of a Differentiable Function [closed]

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 and in this webpage 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?

## closed as off-topic by Nik Weaver, Piotr Hajlasz, abx, Alexandre Eremenko, Jan-Christoph Schlage-PuchtaJan 13 at 11:47

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• Undoubtedly this is in Geometric Measure Theory by Federer. But I think this question belongs on math.stackexchange. – Nik Weaver Jan 11 at 14:49
• It may be hard to find the proof in a textbook, but you can prove it yourself by taking a point $x$ with $\|\nabla f(x) \|\geq K-\varepsilon$ and finding a $y$ with $|f(y)-f(x)| \geq (K-2\varepsilon)|y-x|$ using the definition of differentiability at $x$. – Stanley Snelson Jan 11 at 15:00
• @StanleySnelson: you have to check the reverse inequality too, which is where convexity comes in: any two points are joined by a line segment and you can apply the one-dimensional mean value theorem. – Nik Weaver Jan 11 at 15:54
• You know what, come to think of it, there's a proof of this in a book I just wrote ... – Nik Weaver Jan 11 at 15:59