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Function has at least three fixed points

Source: Romanian District Olympiad 2023 11.3

March 11, 2023
real analysiscontinuityfunction

Problem Statement

Let f:[a,b]→[a,b]f:[a,b]\to[a,b] be a continuous function. It is known that there exist α,β∈(a,b)\alpha,\beta\in (a,b) such that f(α)=af(\alpha)=a and f(β)=bf(\beta)=b. Prove that the function f∘ff\circ f has at least three fixed points.
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