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Continuous functions whose sums of functional powers have distinct monotonies

Source: Romania National Olympiad 2014, Grade XI, Problem 1

March 3, 2019
functionalgebra

Problem Statement

Find all continuous functions f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} that satisfy: (i)id+f \text{(i)}\text{id}+f is nondecreasing (ii) \text{(ii)} There is a natural number m m such that id+f+f2⋯+fm \text{id}+f+f^2\cdots +f^m is nonincreasing.
Here, id \text{id} represents the identity function, and ^ denotes functional power.
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