Lemma: Late source agreement does not force response agreement

There are smooth scalar forcing terms that agree for $t\ge \epsilon$ but whose zero-data responses do not agree there.

Proof. Choose $F\in {C_c}^{\infty }((0,\epsilon );\mathbb{R})$, $F\ge 0$, $F\not\equiv 0$. The sources $F$ and $0$ agree for $t\ge \epsilon$. Their zero-data response difference is

For $t\ge \epsilon$,

In fact $z(t)>0$ for $t\ge \epsilon$. Thus the responses differ after their sources have become equal.