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I got this problem:

Find examples of continuous functions $f:[0,1]\to[0,1]$ that satisfy $\forall x\in[0,1], f(f(x))=f(x)$ other than $f(x)=x$.

I proved that $f([0,1])=\{x\in[0,1]|f(x)=x\}$, But I couldn't find an example of a function other than $f(x)=x$ that satisfies the conditions.

Thanks on any examples.

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2 Answers 2

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The condition $f(f(x)) = f(x)$ is equivalent to $f(y)= y$ for all $y \in $ image of $f$.

Take an $f$ that is not surjective: $f(x) = x$ for $x\le 1/2$ and $f(x) = 1/2$ for $x\ge 1/2$.

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    $\begingroup$ Any retraction of $[0,1]$ onto a closed sub-interval. $\endgroup$
    – orangeskid
    Sep 28, 2014 at 10:08
  • $\begingroup$ Nice answer, by the way can you think of a subjection that satisfies the conditions? $\endgroup$
    – MathNerd
    Sep 28, 2014 at 10:12
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    $\begingroup$ $f$ is surjection means image of $f = [0,1]$. But we have the condition $f(y) = y$ for $y$ in image. Therefore if $f$ surjection then $f(x) = x$ for all $x \in [0,1]$. $\endgroup$
    – orangeskid
    Sep 28, 2014 at 10:19
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    $\begingroup$ Any retraction of $[0,1]$ onto a closed sub-interval. Pick the image $[\alpha ,\beta] \subset [0,1]$. On $[\alpha ,\beta]$ $f$ is the identity. Then flap the arms $[0,\alpha]$ and $[\beta, 1]$ into $[\alpha, \beta]$ (lots of freedom here). $\endgroup$
    – orangeskid
    Sep 28, 2014 at 10:19
  • $\begingroup$ No worries. Try sketching some graphs of such functions. $\endgroup$
    – orangeskid
    Sep 28, 2014 at 10:27
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I was thinking at first that you meant F(f(x))=x (ie involutary; or a function that is its own inverse)

So I was going to mention F(x)=1-x as a possible candidate that is bi=jective nd continuous f:[0,1]\to[0,1]; until I read that you meant f(f(x))=f(x).

It would be interesting to note whether there are any functional equations that uniquely specify f(x)=x without explicitly mentioned a continuity (or continuity like condition); or the the function, being almost already in some sense itself F(F(x))=f(x), specified (at least given it being invertible).

I was wondering, whether if F(x)=x can be uniquely specified by 1.f(f(x))=x, 2.Where F is a one -one function from F:[0.1] to[0,1] closed and bounded. 3.That is Mononotoncally Strictly Increasing in ;F(1)=1; F(0)=0, the end points of co-domain, / max points of domain, codomain etc being specified

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