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This question already has an answer here:

Be $f:[0,1]\longrightarrow[0,1]$ a continuous function, prove that exists $x\in[0,1]$ so that $f(x)=x$ .

I am studying mathematical analysis in functions of one variable, and looking through my notes I can't find any theorem or proposition to help me prove it.

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marked as duplicate by Andrew D. Hwang, Community Jan 30 '17 at 12:45

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    $\begingroup$ Apply IVT on $f(x) - x$. $\endgroup$ – Paramanand Singh Jan 30 '17 at 11:27
  • $\begingroup$ Banach theorem.. $\endgroup$ – josf Jan 30 '17 at 11:53
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Hint.

Consider the function

$$g(x)=f(x)-x$$

and use the intermediate value theorem (this is possible because $f$ is continuous) to prove that there exists $x\in [0,1]$ such that $g(x)=0$.

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