Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying $f(f(x)) =f(x)$ then which one is correct

Question

As stated in the title; Let $f: \mathbb R \to \mathbb R $ be a continuous function satisfying $f(f(x))$ = $f(x)$ then

(a) $f$ must be constant

(b) $f(x) = x$ for all $x$ in range of $f$

(c) $f$ must be a non constant polynomial

(d) There is no such function

By randomly trying different functions I discovered that $f(x) = x$ and $f(x) = 1-x$ satisfy given property . So using this option (c) seems to be correct.

But , My question is that How can I make sure that these are the only functions that hold this property ? and if there are any other function (other than these two) then how should I find them .

Thank you

Answer 1

The answer should be $b$. Suppose not. That means there exists $f$ and there exists $y_0$ such that we have some $x_0$ with $f(x_0)=y_0$ and we have $f(y_0)\neq y_0$.

Then, $f(x_0)=y_0$, but $f(f(x_0))=f(y_0)\neq y_0$, which contradicts the assumtion that $f(f(x))=f(x)$ for all $x$.

Answer 2

If $x$ is in the range of $f$ means $x=f(y)$ for some $y$. Since $f(f(y))=f(y)$ you have $f(x)=x$.

Just to see how arbitrary your function may be I give the following example satisfying your conditions:

$$f(x)=\left\{ \begin{array}{cc} x,&|x|\leq 1\\ \sin(\pi x/2),& |x|>1 \end{array}\right. $$