Some remarks concerning monotone and continuous restrictions of real-valued functions. (English) Zbl 1282.26004
Summary: We consider those restrictions of real-valued functions, which have certain nice properties, e.g. continuity or monotonicity. We prove the non-existence of restrictions of such a kind in concrete situations and show close connections of this topic with some classical examples of sets and functions in real analysis (Luzin sets, Sierpiński sets, continuous nowhere approximately differentiable functions, Sierpiński-Zygmund functions, etc.).
MSC:
26A15 | Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable |
26A48 | Monotonic functions, generalizations |
26A21 | Classification of real functions; Baire classification of sets and functions |