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Thread: continuous function, point in set

  1. #1
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    continuous function, point in set

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

    This looks a lot like Intermediate Value Theorem but with the condition that $\displaystyle f(x)=x$. The condition that $\displaystyle f(x)=x$ is tricky for me, that there is an $\displaystyle x \in [a,b]$ that satisfies this. Thanks for any help.
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  2. #2
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    Suppose that $\displaystyle g(x)=f(x)-x$ is $\displaystyle g$ continuous? WHY?
    Is this true, $\displaystyle g(a)\geqslant 0\;\&\; g(b)\leqslant 0$? If so, then WHY?
    Now use the Intermediate Value Theorem .
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