\(C^{\gamma}\)-property and approximately differentiable functions (Q581690)

We say that a transformation \(f\) from \(R^n\) to \(R^m\) has \(C^{\gamma}\)-property if for every ball \(B\) in \(R^n\) and for each \(\varepsilon >0\) there exists a compact set \(K\subset B\) and the transformation \(g\in \mathrm{lip}^{\gamma}(R^n,R^m)\) such that \(mK >mB- \varepsilon\) and \(f| K=g| K\) (here \(m\) is the \(n\)-dimensional Lebesgue measure). The paper gives three characterizations of the property \(C^{\gamma}\). The main result can be considered as a generalization of theorems of Khintchine, Bernstein and Whitney [see, for example \textit{H. Federer}, ``Geometric measure theory.'' [Berlin etc.: Springer Verlag (1969; Zbl 0176.00801), p. 228].