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|f'(x)|≠1, f has a fixed point and f(b)=1-b

Source: VJIMC 2012 1.1

May 31, 2021
functioncalculus

Problem Statement

Let f:[0,1]→[0,1]f:[0,1]\to[0,1] be a differentiable function such that ∣f′(x)∣≠1|f'(x)|\ne1 for all x∈[0,1]x\in[0,1]. Prove that there exist unique α,β∈[0,1]\alpha,\beta\in[0,1] such that f(α)=αf(\alpha)=\alpha and f(β)=1−βf(\beta)=1-\beta.
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