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Let $f$ be a continuous function with domain $[0,1]$ and range $[0,1]$. Prove that $f$ must have a fixed point: that is, there exists some $a \in [0,1]$ such that $f(a) = a$. Is the result still true if the domain and range are $(0,1)$? Why or why not?

Jul 20, 2022

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Suppose f(0) is positive.  Then f(x) must remain above x for all x, but this is not possible at the end because f(x) must be squeezed between 0 and 1.  So f(a) = a for some a.

Jul 20, 2022