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f(3x) = 3f(x) - 4f(x)^3 , continuous at x = 0, prove that |f(x)| \le 1

Source: 1985 Polish MO Finals p3

January 21, 2020
functioncontinuousinequalities

Problem Statement

The function f:R→Rf : R \to R satisfies f(3x)=3f(x)−4f(x)3f(3x) = 3f(x) - 4f(x)^3 for all real xx and is continuous at x=0x = 0. Show that ∣f(x)∣≤1|f(x)| \le 1 for all xx.
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