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Show that 0 < f(x)-x < c

Source: IMO Longlist 1983

September 9, 2010
algebrafunctional equationcontinuous functionIMO Shortlist

Problem Statement

Let f:[0,1]→Rf : [0, 1] \to \mathbb R be continuous and satisfy: \begin{cases}bf(2x) = f(x), &\mbox{ if } 0 \leq x \leq 1/2,\\ f(x) = b + (1 - b)f(2x - 1), &\mbox{ if } 1/2 \leq x \leq 1,\end{cases} where b=1+c2+cb = \frac{1+c}{2+c}, c>0c > 0. Show that 0<f(x)−x<c0 < f(x)-x < c for every x,0<x<1.x, 0 < x < 1.
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