TITLE: Continuous $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$
QUESTION [8 upvotes]: What are the continuous functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived?
$$f(2x)=f(x-\frac{1}{4})+f(x+\frac{1}{4})\,\,\,\,\,\,\,\,\text{for }x \in (-\frac{1}{4},\frac{1}{4})$$
I think this could be interesting because in addition to the obvious solutions $f(x)=zx$, this is also satisfied by $f(x)=\ln (2\cos(\pi x))$.

REPLY [4 votes]: example 
Using Mike's method, with $f_0(x) = x^2$

Here is an unbounded one, $f_0(x) = \log(4x+2)$
 
So now, what about differentiable, or real-analytic solutions?
Or even solutions continuous at both endpoints.