Show that if $f\in L^1_{loc}(\mathbb{R})$ and $f(x)=\frac{1}{2r}\int_{x-r}^{x+r}f(t)dt$ for all $x\in\mathbb{R},r>0$ then $f\in C^{\infty}$

Question

I'm trying to show that if $f:\mathbb{R}\to\mathbb{R}$ is Lebesgue-integrable, has finite integral over intervals (i.e. $f\in L^1_{loc}$) and satisfies $$f(x) = \frac{1}{2r}\int_{x-r}^{x+r}f(t)dt,\quad\forall x\in\mathbb{R},\forall r>0,$$ then $f\in C^{\infty}$.

The problem also suggests to first prove that $f$ is continuous and use the Fundamental Theorem of Calculus.

I conjecture that $f$ must be of the form $f(x) = ax + b$, but I haven't been able to prove it.