continuous function, point in set

**sophia782** · March 3rd 2009, 02:36 PM

Let $f: [a,b] \rightarrow [a,b]$ be a continuous function. Prove that there is at least one point $x \in [a,b]$ so that $f(x)=x$ .

This looks a lot like Intermediate Value Theorem but with the condition that $f(x)=x$ . The condition that $f(x)=x$ is tricky for me, that there is an $x \in [a,b]$ that satisfies this. Thanks for any help.

**Plato** · March 3rd 2009, 03:11 PM

Suppose that $g(x)=f(x)-x$ is $g$ continuous? WHY?
Is this true, $g(a)\geqslant 0\;\&\; g(b)\leqslant 0$ ? If so, then WHY?
Now use the Intermediate Value Theorem .

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