Głąb, Szymon Local and global monotonicity. (English) Zbl 1044.26008 Real Anal. Exch. 27(2001-2002), No. 2, 765-771 (2002). Summary: We give characterizations of sets \(E\subset [0,1]\) for which the local monotonicity of each function \(f:[0,1]\to\mathbb{R}\) from a given class \({\mathcal F}\), at all points \(x\in E\), implies the global monotonicity of \(f\) on \([0,1]\). We consider as \({\mathcal F}\) – the families of continuous functions, differentiable functions, absolutely continuous functions, functions of class \(C^n\) \((n=1,2,\dots,\infty)\), real analytic functions and polynomials. Cited in 1 Document MSC: 26A48 Monotonic functions, generalizations 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26A46 Absolutely continuous real functions in one variable 26A42 Integrals of Riemann, Stieltjes and Lebesgue type Keywords:local monotonicity; Cantor set; Cantor-Bendixson derivative; condition N of Luzin; global monotonicity PDFBibTeX XMLCite \textit{S. Głąb}, Real Anal. Exch. 27, No. 2, 765--771 (2002; Zbl 1044.26008) Full Text: DOI