\(C^ r\)-property and approximately differentiable functions (Q1177818)

This is a continuation of a previous paper by the same author [Mat. Zametki 44, No. 5, 645-659 (1988; Zbl 0689.26008)]. The main result is the characterization of mappings having \(C^ r\)-property in terms of \(V^ r_ x(f;\rho)\) and \(\text{dist}(0,F\rho(x))\). Recall that a mapping \(f:\mathbb{R}^ n\to\mathbb{R}^ m\) has \(C^ r\) property if for every ball \(B\subset\mathbb{R}^ n\) and every \(\varepsilon>0\) there exists a compact set \(K\subset B\) and a mapping \(g\in C^ r(\mathbb{R}^ n,\mathbb{R}^ m)\) such that \(m(K)>m(B)-\varepsilon\) and \(f| K=g| K\).