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Math Help - continuous function, point in set

  1. #1
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    continuous function, point in set

    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.
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  2. #2
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    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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