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f(x)f(f(x)) = 1 , f(1000) = 999, continuous, f(500)?

Source: Polish second round 1993 p6

January 19, 2020
functioncontinuous functionalgebrafunctional equation

Problem Statement

A continuous function f:Rā†’Rf : R \to R satisfies the conditions f(1000)=999f(1000) = 999 and f(x)f(f(x))=1f(x)f(f(x)) = 1 for all real xx. Determine f(500)f(500).
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