To check if a system is memory less or not

Question

I need to check whether the system $R$: $f\to g$ given by $$g(t) = (Rf)(t) = \int_{-\infty}^{t}f(\tau)e^{-(t-\tau)}d\tau$$ is memory less or not

Intuitively, since we are summing from minus infinity to present time, the system seems to depend on previous states and hence should not be memory less. But I am unable to prove it mathematically. Request guide.

Answer 1

A memoryless system has an impulse response of the form $K\delta(t)$ , where $K$ is a constant. That is, the output at time $t$ only depends on the input at time $t$.

You can see that the impulse response of the given system is $$h(t)=e^{-t}u(t).$$