MathDB
Hard problem (i think)

Source: Romanian MO

March 8, 2006
functioninductionreal analysisreal analysis unsolved

Problem Statement

Let two bijective and continuous functionsf,g:R→Rf,g: \mathbb{R}\to\mathbb{R} such that : (f∘g−1)(x)+(g∘f−1)(x)=2x\left(f\circ g^{-1}\right)(x)+\left(g\circ f^{-1}\right)(x)=2x for any real xx. Show that If we have a value x0∈Rx_{0}\in\mathbb{R} such that f(x0)=g(x0)f(x_{0})=g(x_{0}), then f=gf=g.
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