Let X be connected, and assume a homeomorphism A:X→X exists such that A∘A(x)=x for all x∈X. Prove that, for every continuous function f:X→ℝ, there exists x∈X such that f(x)=f(A(x)).

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- March 18th 2009, 08:20 PMAndreametIVT
Let X be connected, and assume a homeomorphism A:X→X exists such that A∘A(x)=x for all x∈X. Prove that, for every continuous function f:X→ℝ, there exists x∈X such that f(x)=f(A(x)).