Request
I need to prove or to refute the the claim below, I tried to prove the claim but run into troubles with the $\Rightarrow$ side of the proof.
Given
$f$ is a continuous function $[0,\infty) \to \mathbb R $, then $ \forall t\geq0: (f * f)_t = 0 $ iff $f\equiv0$.
My solution
The $\Leftarrow$ part of the proof is trivial, but how to prove the other direction? Maybe even my line of thought is wrong and i could refute the claim?
Note: We didn't prove that the transform is unique hence i cannot just use inverse Laplace transform and infer that $f(t)=0$.