What Are the Possible Functions That Satisfy f(f(x)=f(x) on [0,1]?

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The discussion centers on identifying functions that satisfy the equation f(f(x)) = f(x) on the interval [0,1]. The established solutions include f(x) = x and f(x) = c, where c is a constant. Additionally, participants propose piecewise functions such as f(x) = 1/2 for x ≤ 1/2 and f(x) = 1 for x > 1/2, as well as f(x) = floor(x + 1/2), which also meet the criteria under specific conditions. The conversation highlights the complexity of the problem, especially when considering discontinuous functions or those defined differently over rationals and irrationals.

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Dazedandconfu
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aite, here goes f(f(x)=f(x), find all functions that satisfy this on [0,1], i know f(x)=x and f(x)=c satisfy this equation, but I am looking for something more outlandish, feel free to make it discontinuous, define it separately over the rationals and irrationals, or any sort of crazy function, i have no clue about the answer, just thought it was an interesting question
 
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I don't think there are any functions apart from those you mentioned that satisfy that identity. Assume f^(-1), which denotes the inverse of f(x), exists. Then taking f^(-1) both sides, gives f(x)=x. Setting, x=c gives the other case. Although I wouldn't be surprised if there are some brilliant functions that satisfy this.
 
Charles49 is right if we assume f is invertible, and f(x)=x is the only solution for it.
If not, then we can get some crazy functions like
f(x) = 1/2 if x <= 1/2
f(x) = 1 if x > 1/2
This will also satisfy the question.

(maybe this might be true for cantor function?)

I think this question is quite deep if we don't have any restrictions on f,
because we can have this too:
f(x) = floor(x+1/2), and it satisfies the problem too. (nvm for this, it only works for [0,1) )
 

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