Consider the function $f : \mathbb{R} \to [0,1]$ with $$ f(x) = \begin{cases} -1 & x \le -1 \\ +1 & x \ge +1 \\ \frac{f(\frac32 (x-\frac13)) + f(\frac32 (x+\frac13))}{2} & -1 \le x \le +1\,. \end{cases} $$ So $f$ is the average of two affine transformations of itself. The picture shows the graph of $f$ (in red) and the two affine transformations from the definition (in blue). What is known about such functions? Is $f(0.5) \approx 0.694064$ rational?