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

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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.

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

• Apply IVT on $f(x) - x$. – Paramanand Singh Jan 30 '17 at 11:27
$$g(x)=f(x)-x$$
and use the intermediate value theorem (this is possible because $f$ is continuous) to prove that there exists $x\in [0,1]$ such that $g(x)=0$.