Points of continuity of a function and points of existence of finite and infinite derivative (Q1059725)

In his previous papers [e.g. Sib. Mat. Zh. 18, 806-820 (1977; Zbl 0369.26004)] the author derived some topological properties of the sets \(Q_ f\), \(M_ f\), \(N_ f\) (subsets of the real line R and determined by a function \(f: R\to R\) as stated below) which are the set of all discontinuity points of f, the set of all points \(x\in R\) with \(f'(x)=\infty\), and the set of all points \(x\in R\) where the derivative f'(x) does not exist, respectively. According to the present carefully written paper, the mentioned properties completely characterize these triples Q, M, N, i.e., they are sufficient for the existence of a function \(f: R\to R\) with \(Q=Q_ f\), \(M=M_ f\), \(N=N_ f\). The author also succeed in determining necessary and sufficient conditions on a subset \(K\subset R\) for the existence of a function \(f: R\to R\) of type \(\beta\) (i.e., the derivative f'(x) exists at all continuity points x) such that the (finite or infinite) derivative f'(x) exists exactly at \(x\in K\).