Reference Request: Smooth Arzela-Ascoli Theorem

Question

I am using this result usually tied to the Arzela-Ascoli theorem:

Let F be a subset of C(X) where X is a convex set in $R^n$ . Suppose that each function in F is differentiable and there is a uniform bound on the partial derivatives of these functions in F. Then F is equicontinuous.

But I cannot find a reference which I can cite for my project...would anyone know where I can find this in a book or paper? Every version I find is either more general or less than this exact version...

Answer 1

This result is not any kind of Arzela-Ascoli theorem (which is a statement about compactness). Rather, it is a form of the Mean Value Theorem, or perhaps Mean Value Inequality (since the relation $f(a)-f(b) = f'(c)(a-b)$ becomes an inequality in higher dimensions):
$$ |f(a)-f(b)|\le \sup_\Omega \|\nabla u\| \ \|a-b\| $$ where $\Omega$ is a convex set containing $a, b$.

It can be found in Rudin's Principles of Mathematical Analysis (Theorem 9.19, page 218 of the 3rd edition), and I'd expect it to be in good multivariable calculus textbooks like Apostol's.