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Local and global monotonicity. (English) Zbl 1044.26008

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.

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
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