Solve $f(x)=\int_{x-1}^{x+1} f(t) \text{d}t$.

When $f(x)$ is a function, looks like the only solution is $f(x)=0$. But what if we allow distributions, such as DiracDelta?

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$.

When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?