Q:

Find all bijections $\,\,f:[0,1]\rightarrow[0,1],\,$ which satisfy $\,\,f\big(2x-f(x)\big)=x$.

A friend of mine gave me the following problem: 
Find all functions $f:[0,1]\to[0,1]$, which are one-to-one and onto and satisfy the following functional relation:
$$
f\big(2x-f(x)\big)=x, \tag{1}
$$
for all $x\in [0,1]$.
Clearly, the identity function $f(x)=x$ is one such function.
Also, as $f$ is a bijection $f^{-1}$ exists, and by $(1)$ we have
$$
f^{-1}(x)=2x-f(x), \tag{2}
$$
but I have no idea that how should I continue. It will be great if someone can give me some hints. 
Thanks in advance.

A:

In the formulation of the question the function $f$ is one-to-one and onto. Below two proofs are presented. The first one requires the one-to-one and assumption, while the second one does not.

First answer requiring that $f$ is one-to-one.
Clearly, $f$ is also onto, as an arbitrary $x\in[0,1]$, is the image of $2x-f(x)$, and since
$f$ is one-to-one and onto, $f$ possesses an inverse $f^{-1}:[0,1]\to [0,1]$. 
The functional relation $f\big(2x-f(x)\big)=x$, implies
$$
f(x)-x=x-f^{-1}(x). \tag{1}
$$
We shall show that $f(x)=x$. Assume not. Then 
$$
f(x_0)-x_0\ne 0,
$$
for some $x_0\in(0,1)$. Let's assume that $f(x_0)-x_0=a>0$. Then $(1)$ implies that
$$
f(x_0)-x_0=x_0-f^{-1}(x_0)=f^{-1}(x_0)-f^{-2}(x_0)=\cdots=f^{-k}(x_0)-f^{-(k+1)}(x_0),
$$
for every $k\in\mathbb N$, where $f^{-k}$ is $f^{-1}\circ\cdots\circ f^{-1}$ $k$ times.
But this means that 
$$
f^{-1}(x_0)=x_0-a,\,\,f^{-2}(x_0)=x_0-2a,\ldots,f^{-k}(x_0)=x_0-ka,
$$
which means that $\lim_{k\to\infty}f^{-k}(x_0)=-\infty$. A contradiction, 
$f^{-k}(x_0)\in [0,1]$.
We would reach to a contradiction even if we had assumed that $a<0$. 

Second answer not requiring that $f$ is one-to-one.
As $f\big(2x - f(x)\big) = x,\,$ for all $x \in \left[ {0,1} \right]$, then
$$
0 \le 2x - f(x) \le 1 \quad \Longrightarrow\quad 2x - 1 \le f(x) \le 2x,\,\,\,
\text{for all $x \in \left[ {0,1} \right]$.}
$$
Replacing in the above $x$ with  $2x - f(x)$ we obtain
\begin{align}
2\big( {2x - f(x)} \big) - 1 \le f\big( {2x - f(x)} \big) \le 2\big( {2x - f(x)} \big),
\end{align}
or
\begin{align}
2\big( {2x - f(x)} \big) - 1 \le x  \le 2\big( {2x - f(x)} \big)
\end{align}
which implies that
$$
\frac{3x-1}{2}\le f(x) \le \frac{3x}{2}.
$$
Repetition of this process produces the following inequalities
$$
x+\frac{x-1}{n}\le f(x)\le x+\frac{x}{n},\,\,\,
\text{for all $x \in \left[ {0,1} \right]\,\,$ and $\,\,n\in\mathbb N$,}
$$
and therefore
$$
f(x)=x ,\,\,\,
\text{for all $x \in \left[ {0,1} \right]$.}
$$

A:

OK guys, thanks to @woso's suggestion I found an answer. Let me share it with you.

First Answer
By $(2)$ we have
$$f(x)-x=x-f^{-1}(x)\tag{3}$$
Now let $x_0\in[0,1]$ be arbitrary. We define the sequence $(x_n)_{n=1}^{\infty}$ as follows
$$x_n=f(x_{n-1})\tag{4}$$
for all $n\ge1$. From $(3)$ and $(4)$ we have
$$\begin{align*}x_n-x_{n-1}&=x_{n-1}-x_{n-2}\\ &= x_{n-2}-x_{n-3}\\ &= x_{n-3}-x_{n-4}\\&\vdots\\ &= x_{1}-x_{0}\tag{5}\end{align*}$$
Now from $(5)$ we have
$$x_n-x_0=\sum_{k=1}^{n}{(x_k-x_{k-1})}=\sum_{k=1}^{n}{(x_1-x_0)}=n(x_1-x_0)\tag{6}$$
Now since $x_n,x_0\in[0,1]$  so by $(6)$ we have
$$n|x_1-x_0|=|x_n-x_0|\le 1\tag{7}$$
so by $(7)$ for alll $n\in\Bbb{N}$ we have $|x_1-x_0|\le \frac{1}{n}$, but it's true only if $|x_1-x_0|=0$, so finaly
$$f(x_0)=x_1=x_0$$
since $x_0\in[0,1]$ was arbitrary so $f(x)=x$ for all $x\in[0,1]$ and it's the only solution.

Second Answer
We can show that $f(x)=x$ is the only solution even without considering $1$-$1$ and onto conditions.
For all $x\in[0,1]$ consider $g(x)=x-f(x)$. Let $x_0\in[0,1]$ be arbitrary, then $g(x_0)=x_0-f(x_0)$ and by assumption $x_1=x_0+g(x_0)=2x_0-f(x_0)\in[0,1]$, also we can see that $g(x_1)=g(x_0)$, the same process shows that $x_2=x_1+g(x_1)\in[0,1]$ and $g(x_2)=g(x_1)=g(x_0)$,so the sequence $(x_n)$ defined by $x_n=x_{n-1}+g(x_{n-1})$ has these properties:
1. for all $n\ge0$, $x_n\in[0,1]$;
2. for all $n\ge1$, $g(x_n)=g(x_0)$.
so $x_n-x_{n-1}=g(x_{n-1})=g(x_0)$, and
$$x_n-x_0=\sum_{k=1}^{n}{(x_k-x_{k-1})}=\sum_{k=1}^{n}{g(x_0)}=ng(x_0) \tag{*}$$
Now since $x_n,x_0\in[0,1]$ so by $(*)$
$$n|g(x_0)|=|x_n-x_0|\le 1\tag{**}$$
so by $(**)$ for all $n\in\Bbb{N}$ we have $|g(x_0)|\le\frac{1}{n}$, but it's true only if $g(x_0)=0$, so finally
$$f(x_0)=x_0-g(x_0)=x_0$$
since $x_0\in[0,1]$ was arbitrary so $f(x)=x$ for all $x\in[0,1]$and it's the only solution.

If you find something wrong with it, please add a comment.

