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Two continuous functions

Source: Iranian National Olympiad (3rd Round) 2008

September 12, 2008
functioncomplex analysiscomplex analysis unsolved

Problem Statement

Let g,f:C⟶C g,f: \mathbb C\longrightarrow\mathbb C be two continuous functions such that for each z≠0 z\neq 0, g(z)\equal{}f(\frac1z). Prove that there is a z∈C z\in\mathbb C such that f(\frac1z)\equal{}f(\minus{}\bar z)
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