Local and Global Monotonicity

Abstract

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 $C^n$ ($n=1,2,...,\infty$), real analytic functions and polynomials.