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

Question

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.

Answer 1

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 $g(x)=0$.