Functional equation which should have $f(x)=x$ as the unique solution

Question

I'd like to solve the following functional equation: \begin{equation} 2f(x)+f(1-2x)=1 \end{equation}

for $x \in [0,1/2]$. I also know that $f(1/2)=1/2$.

This equation arises from a problem where I know I should get $f(x)=x$ as the unique solution. I can't find anything online, the main problem being that $1-2x$ is not involutive.

Answer 1

Define $f(x)=x$ if $x$ is rational and $f(x)=1/3$ if $x$ is irrational. Then we have

$$2f(x)+f(1-2x)=1$$

for all $x$.

Answer 2

I will show the following conditions are enough to uniquely define the function $f:\mathbb{R}\to\mathbb{R}$ $$\forall{x\in\mathbb{R}} \quad 2f(x)+f(1-2x)=1$$ $$\forall{x\in\mathbb{R}} \quad f\text{ is continuous at }x$$ $$f \text{ is known for some } x \ne \frac{1}{3}$$ First, lets find linear solutions to the first condition. $$f(x)=ax+b \quad\quad 2f(x)+f(1-2x)=1$$ $$2(ax+b)+a(1-2x)+b=1$$ $$2ax+2b+a-2ax+b=1$$ $$3b+a=1 \quad\quad b=\frac{1-a}{3}$$ $$f(x)=ax-\frac{a}{3}+\frac{1}{3}$$ $$f(x)=a\left(x-\frac{1}{3}\right)+\frac{1}{3}$$ Knowing $f$ at one $x$ also defines $f$ at many other $x$. The entire set of $x$ that $f$ must be defined on for one seed $x_0$ can be given by all the terms of the sequence: $$x_{n+1}=1-2x_n$$ The sequence is valid for all $n \in \mathbb{Z}$. Each term of this sequence can be given by the explicit formula: $$x_n=(-2)^n\left(x_0-\frac{1}{3}\right)+\frac{1}{3}$$ $$\text{Observe that:}$$ $$\lim_{n \to -\infty}x_n=\frac{1}{3}$$ This formula can be derived by observing that the sequence is a special case of compositions of: $$ax+b=a^1x+b(1)$$ $$a(ax+b)+b=a^2x+b(a+1)$$ $$a(a(ax+b)+b)+b=a^3x+b(a^2+a+1)$$ $$a(a(a(ax+b)+b)+b)+b=a^4x+b(a^3+a^2+a+1)$$ The remaining summation is a partial geometric series. The resulting explicit formula can be shown to match the sequence for all $n \in \mathbb{Z}$ using induction given that: $$a=-2 \quad\quad b=1$$ Every solution must pass through the point $\left(\frac{1}{3}, \frac{1}{3}\right)$ as shown by plugging $x=\frac{1}{3}$ directly into the functional equation. Consider the seed point of condition three: $(x_0, y_0)$. The linear function passing through both points is one possible solution. The linear solution along with additional points the seed generates ensures: $$\forall n \in \mathbb{Z} \quad\quad f\left((-2)^n\left(x_0-\frac{1}{3}\right)+\frac{1}{3}\right)=(-2)^n\left(x_0-\frac{1}{3}\right)\left(\frac{y_0-\frac{1}{3}}{x_0-\frac{1}{3}}\right)+\frac{1}{3}$$ Consider seed intervals of $x$ that uniquely define $f$. Clearly knowing the value of $f$ over the domain $(-\infty, \infty)$ uniquely defines $f$. We will perform a series of reductions to try to get a seed interval of minimum length. $\left[\frac{1}{3}, \infty\right)$ works because $1-2x$ applied to the interval yields $\left(-\infty, \frac{1}{3}\right]$ whose union is all reals. We will use $1-2x$ composed with itself $(4x-1)$ and its inverse $\left(\frac{x+1}{4}\right)$ to further reduce the interval. Consider the seed interval $[1, 3)$. First use the $4x-1$ mapping on the interval repeatedly: $$[1, 3)\to[3, 11)\to[11, 43)\to[43, 171)\to\dots\infty$$ And now the $\frac{x+1}{4}$ mapping: $$[1, 3)\to[0.5, 1)\to[0.375, 0.5)\to[0.34375, 0.375)\to\dots\frac{1}{3}$$ The union of all these intervals is $(\frac{1}{3}, \infty)$. We don't need to worry about the exclusion of $x=\frac{1}{3}$ because $f$ is known to be $\frac{1}{3}$ there regardless. If we reduced from $\left(-\infty, \frac{1}{3}\right]$ instead of $\left[\frac{1}{3}, \infty\right)$, a similar result would be obtained. Let $P(I)$ be a statement expressing the unique defining of $f$ from the seed interval $I$. This process shows: $$\forall x\ne\frac{1}{3} \quad P[x, 4x-1)$$ If $P(A)$ and $A \subseteq B$, then $P(B)$. We may expand the seed interval in whatever way we choose; making smaller intervals is the only part requiring reasoning. Expand the intervals in the statement to: $$\forall \epsilon>0 \quad P\left(-\epsilon+\frac{1}{3}, \frac{1}{3}+\epsilon\right)$$ For a given seed point, we must show the only continuous solution is: $$f(x)=\left(x-\frac{1}{3}\right)\left(\frac{y_0-\frac{1}{3}}{x_0-\frac{1}{3}}\right)+\frac{1}{3}$$ I am not entirely sure about the rest of the proof. It seems intuitive enough, but I have been unable to formalize certain parts sufficiently. I imagine the epsilon delta definition of limit could be used to complete the proof with rigor. I will try to finish it later.
Let the line defined above for a given seed point be called $L$. Let the point $\left(\frac{1}{3}, \frac{1}{3}\right)$ be denoted as $P$. Assume there exists a continuous solution not equal to $L$. Let this solution curve be denoted as $C$. $C$ intersects with $L$ at $P$. If $C$ were to match $L$ for any finite distance around $P$, then $C$ would match $L$ everywhere due to the seed interval statement, so $C$ must have at least one point infinitely close to $P$ that is not in $L$. Since the seed point belongs to $C$, there are also points in $C$ infinitely close to $P$ that are in $L$ due to generated points from the seed converging to $P$ on $L$. I believe the multiple ways $C$ must approach $P$ in order to be different from $L$ ensure $C$ is discontinuous somewhere else when the infinitesimal differences around $P$ are blown up, but I do not really know yet.