TITLE: There exists x on closed interval such that f(x)=x
QUESTION [8 upvotes]: If $f$ is a continuous function on a closed interval, how can I show that there exists some $x$ on $f$ that $f(x)=x$?
I know it will require the Intermediate Value Theorem. Initially I thought of letting $g(x)=f(x)-x$, and showing that there exists $x$ such that $g(x)=0$ implying $f(x)=x$, but don't have any information about $f$ on either end of the interval.

REPLY [3 votes]: Since $f(0) \in [0,1]$, then $g(0)=f(0)-0\ge 0$; and Since $f(0) \in [0,1]$, then $g(1)=f(1)-1\le 0.$
Hence $g(0)\cdot g(1) \le 0$.