Doubts about the stability of a system.

Question

For the following dynamic system: $$\dot{x} = Ax + f(t)$$ $f(t)$ is piecewise continuous and differentiable. If for any bounded initial value, we know $\lim_{t\rightarrow \infty}x(t) = 0$.We can get $A$ is Hurwitz, but we can’t seem to get $\lim_{t\rightarrow \infty} f(t) = 0$, in the last question I asked, we can see that the limit of $f(t)$ may not exist. Therefore, I have a doubt, how to restrict the condition of $f(t)$ so that we can get the necessary and sufficient condition of $\lim_{t\rightarrow \infty }x(t) = 0$. In other words, what conditions does $f(t)$ meet to make the following conclusions true? $$\lim_{t\rightarrow \infty }x(t) = 0 \Longleftrightarrow A\quad\mathrm{is}\quad \mathrm{Hurwitz} \quad \&\& \quad \lim_{t\rightarrow \infty} f(t) = 0$$