Finding the function

Question

I would appreciate if somebody could help me with the following problem:

Q: $f(x):$ conti-function and $f(2x)-f(x)=x^3$

find $f(x)=?$

Answer 1

One more or less obvious solution is $f(x)=\frac17x^3$ (try an ansatz $ax^3$ and determine $a$).

If $f$ is an arbitrary soluiton, let $g(x)=f(x)-\frac17x^3$. Then $g$ is also continuous and $g(2x)-g(x)=0$. Then $g$ is constant as for any $x$ we find $g(x)=g(2^{-n}x)\to g(0)$ as $n\to\infty$. Therefore the most general solution to the original problem is $$ f(x)=\frac17x^3+c.$$