Theorems on the smoothness of the right inverse (Q1109907)

Let K be a compact subset in \(R^ n\) and let f be a Lipschitz mapping from K into \(R^ n\). Then there exists a countable family of mappings \(g\nu\) from f(K) into K (\(\nu\in N)\), approximately differentiable on f(K), such that for each \(y\in f(K)\) we have \(g_{\nu}(y)\in f^{- 1}(y)\quad (\nu \in N)\) and the set \(\{g_{\nu}(y)\}_{\nu \in N}\) is everywhere dense in \(f^{-1}(y)\). Moreover, each mapping g, measurable on f(K), such that \(f\circ g=id_{f(K)}\) satisfies the following condition. For each \(\epsilon >0\) there exists a compactum \(C\subset f(K)\) such that \(\mu C>\mu f(K)-\epsilon\) and \(g|_ C\) is Lipschitzian (\(\mu\) is the Lebesgue measure on \(R^ n)\).