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Does there exist continuous functios $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ \ $g\big((0,1)\big)$ ? The problem I am having is that since $(0,1)$ is not compact I am not able to tell anything decisive ; I know that the images of $f,g$ must be intervals but only that . What would happen if I replace $(0,1)$ by $[0,1)$ ? ( In general I am also interested in open bounded intervals $(a,b)$ and bounded half-open intervals $[a,b)$ ) . Please help . Thanks in advance .

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  • $\begingroup$ I think you should think about connectedness instead. $\endgroup$ – IAmNoOne Dec 17 '14 at 5:00
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$f(x)=2x(1-x)$

$g(x)=1-\frac{x}{2}$

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  • $\begingroup$ Fernandez : What about $[0,1) \to [0,1)$ then ? $\endgroup$ – user123733 Dec 17 '14 at 5:17
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    $\begingroup$ $f(x)=\frac{x}{2},g(x)=\frac{1}{2}+\frac{x}{2}$ $\endgroup$ – Jorge Dec 17 '14 at 5:23
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You could e.g. have $g((0,1)) = (0,1/2)$ (very easy) and $f((0,1)) = [1/2,1)$ (hint: $f$ attains a minimum...).

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