Selective differentiation of typical continuous functions (Q1068221)

The results of the paper are: For each Lebesgue measurable function f there exists a selection s and a set P of the cardinality c such that f has the selective derivative sf' at every point of P. The subset of all such continuous functions f of the space \(C(<0,1>)\) with supremum norm, that the selective derivative sf' of f exists at most on a set of Lebesgue measure zero and of the first category for every selection s, is a residual set in \(C(<0,1>).\) Let A' be the set of limit points of the set A. For each natural n we define inductively: \(A^{(1)}=A'\) and \(A^{(n+1)}=(A^{(n)})'.\) Let f be a function: \(<0,1>\to R\) and K a subset of \(<0,1>\) which satisfy the following conditions: (i) if \(x\in K\) and f'(x) exists, then \(f'(x)\in (-\infty,\infty),\) (ii) for every \(x\in K\) it holds: \(D_ Lf(x)\cap D_ Rf(x)\neq \emptyset,\) where \(D_ Lf(x)\) \((D_ Rf(x))\) is the set of all left-sided (right-sided) derived numbers at x, (iii) there exists a natural s such that \(K\cap K^{(s)}=\emptyset.\) Then for any function \(g:K\to R\) such that \(g(x)\in D_ Lf(x)\cap D_ Rf(x)\) for each \(x\in K,\) there exists a selection s such that \(sf'(x)\) exists and \(sf'(x)=g(x)\) for each \(x\in K.\)