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.


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

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.

  • 1
    $\begingroup$ 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


Consider the function


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


Not the answer you're looking for? Browse other questions tagged or ask your own question.