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Thread: Continuity Proof

  1. #1
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    Continuity Proof

    Let f(x) be a function that is de fined for all real x and is continuous on its domain.

    Suppose that lim (x-> negative infinity) =infinity and lim (x-> infinity)= negative infinity.

    Show that there is a value of x such that f(f(x)) = x.



    My intuition says that I should use the IVT, but I'm not sure how...any suggestions?
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  2. #2
    Junior Member nimon's Avatar
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    Consider the continuous function $\displaystyle f(x)-x$. Since $\displaystyle f(x)\rightarrow \infty$ as $\displaystyle x \rightarrow -\infty$ there must be some point $\displaystyle y\in\mathbb{R}$ where $\displaystyle f(y)>y$ (you should prove this). Similarly, there must be some point $\displaystyle z\in\mathbb{R}$ where $\displaystyle f(z)<z$. Then since $\displaystyle 0 \in [f(z)-z,f(y)-y]$, there must be some $\displaystyle x\in[z,y]$ such that $\displaystyle f(x)-x = 0$ by the IVT. Then clearly $\displaystyle f(f(x)) = f(x) = x.$

    So you were right about the intermediate value theorem!
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