Let $p$ be a zero of $2^{p+1}-p-2$ other than $-1$ and $0$ (e.g. one is approximately $2.54536493037426+10.7539751752688 i$). Then the real and imaginary parts of $f(x) = x^p$ satisfy the equation for any $0<a<b$.

Let $p$ be a zero of $2^{p+1}-p-2$ other than $-1$ and $0$ (e.g. one is approximately $2.54536493037426+10.7539751752688 i$). Then the real and imaginary parts of $f(x) = x^p$ satisfy the equation. Note that (with $f(0)=0$) $f$ is continuous on $[0,\infty)$ if $\text{Re}(p) > 0$.