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

Question

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.

Apply IVT on $f(x) - x$. – Paramanand Singh Jan 30 '17 at 11:27 — Paramanand Singh, Jan 30 '17 at 11:27

score 5 · Accepted Answer · answered Jan 30 '17 at 11:27

5

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 $g(x)=0$.

answered Jan 30 '17 at 11:27

E. Joseph

15,066

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

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