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Inifinitely Many Increasing Functions

Source: Romanian District Olympiad 2018 - Grade IX - Problem 4

March 10, 2018
functionromania

Problem Statement

Let f:R→Rf:\mathbb{R} \to\mathbb{R} be a function. For every a∈Za\in\mathbb{Z} consider the function fa:R→Rf_a : \mathbb{R} \to\mathbb{R}, fa(x)=(x−a)f(x)f_a(x) = (x - a)f(x). Prove that if there exist infinitely many values a∈Za\in\mathbb{Z} for which the functions faf_a are increasing, then the function ff is monotonic.
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