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This question already has an answer here:

Suppose $f, g: [0,1] \rightarrow [0,1]$ are continuous functions, such that for each $x$ from $[0,1]$ the identity $f(g(x)) = g(f(x))$ is true. Does there always exist $y\in[0, 1]$, such that $f(y) = g(y)$?

That statement seems to be true, but I do not know how to prove it.

Any help will be appreciated.

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marked as duplicate by Paramanand Singh calculus Aug 10 '17 at 15:24

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ @TacNayn Indeed. I deleted my comment accordingly. $\endgroup$ – lulu Aug 10 '17 at 15:03
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    $\begingroup$ I added a proof on the link $\endgroup$ – Jorge Fernández Hidalgo Aug 10 '17 at 15:38
  • $\begingroup$ For those who just want the conclusion, the statement is true, as proven in the linked question. $\endgroup$ – peterwhy Aug 10 '17 at 15:59