On summability of partial derivatives of Lipschitz functions
Let \(X\) be a bounded measurable subset in \(R^k\), a function \(f\:X\to R^k\) and \(\gamma\in(0,1]\). In [Real Anal. Exch. 29, No. 2, 713--728 (2003--2004; Zbl 1094.26009)] the authors showed the importance of the function \(M_{(f,\gamma)}(x)=\limsup_{y\to x}| f(x)-f(y)|/| x-y|^\gamma\), \(x\in X\), in the study of \(\gamma\)-Hölder functions. Here, they show that the summability of \(M_{(f,1)}\) in \(X\) is equivalent with the summability of \(f\) and differentiability of its partial derivatives in \(X\). As a consequence they prove that \(f\) is differentiable almost everywhere if and only if \(f\) can be in a certain way approximated by Lipschitz functions.
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