*Final update: on 11/29/2019: I have worked on this a bit more, and wrote an article summarizing all the main findings. You can read it here.*

This functional equation appears in the following context. Let $\alpha\in[0,1]$ be an irrational number (called *seed*) and consider the sequence $x_n=\{2^n \alpha\}$. Here the brackets represent the fractional part function. In particular, $\lfloor 2x_n\rfloor$ is the $n$-th digit of $\alpha$ in base $2$.

The values $x_n$ are distributed in a certain way due to the *ergodicity* of the underlying process. The density associated with this distribution is the function $f$, and for the immense majority of seeds $\alpha$ that density is uniform on $[0, 1]$, that is, $f(x) = 1, x \in [0, 1]$. Such seeds $\alpha$ producing the uniform density are sometimes called *normal* numbers; their digit distribution is also uniform.

However, the functional equation $f(x) = \frac{1}{2}\Big(f(\frac{x}{2}) + f(\frac{1+x}{2})\Big)$ may have plenty of other solutions. Such solutions are called *non-standard* solutions. Can you find a seed $\alpha$ producing a non-standard solution, with an explicit form for $f$? Maybe a step-wise uniform function? The set of seeds producing non-standard solutions is known to have Lebesgue measure zero, but there are infinitely many such seeds.

All rational seeds $\alpha$ work, but they produce a discrete distribution. Thus their density is of the discrete type. Here however, I am interested in a continuous function $f$, even if it has infinitely many points of discontinuity (that is, a function $f$ continuous almost everywhere: the set of discontinuity points has Lebesgue measure zero.)

**Update**

I am looking for a function $f$ that is a density on $[0, 1]$, so there are additional constraints here: $\int_0^1 f(x)\,dx = 1$ and $f(x) \geq 0$. However, note that if $f$ is a solution, then $cf$ is also a solution regardless of the constant $c$. So any solution can be normalized to integrate to one. Also, $cf+d$ is also a solution ($c, d$ constants).

**Second update**

Below is a density satisfying all the requirements. Actually, the plot below represents its percentile distribution. It was produced with a seed $\alpha$ built as follows: its $n$-th binary digit is $1$ if $\mbox{Rand}(n) < 0.75$, and $0$ otherwise, using a pseudo random number generator. Note that $P._{25} = 0.5$ and corresponds to a dip ($P._{25}$ denotes the $25$-th percentile.) Dips are everywhere, only the big ones are visible. By contrast, the percentile distribution for the uniform case (if you replace $0.75$ by $0.50$ in $\mbox{Rand}(n) < 0.75$) is a straight line, with no dips.

Note: I eventually answered my question, see the second answer.