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Suppose that a function $f$ is continuous on the closed interval $[0,1]$ and that $0\leq f(x)\leq 1$ for every $x \in [0,1]$. Show that there must exist a number $c$ such that $f(c)=c.$

Lucas
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İrem Varol
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1 Answers1

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$Hint$: remember that when you have a continuos function defined on an interval, you can apply the intermediate value theorem. if $0 \leq f(x) \leq 1$ for each $x\in [0, 1]$, what can you say about the function $h: [0, 1] \rightarrow [0, 1]$ defined by $h(x) = f(x) - x$?

javierochomil
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  • why h(x) is equal to 0 ? i am really confused about this theorem :D – İrem Varol Oct 15 '18 at 23:46
  • You should really try to solve this one yourself, it's a classic exercise of a decent calculus course. I'll give you another hint. Do different cases. What happens if $h(0) = 0$ or $h(1) = 0$? What happens if this is not the case? Give the intermediate value theorem another look. – javierochomil Oct 15 '18 at 23:52
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    thank you :)))) – İrem Varol Oct 16 '18 at 00:02