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 $f(x)=x$

Question

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.

Answer 1

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 $x\ge 1/2$.

Answer 2

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