I have a question about the convolution integral resulting due to an inverse Laplace transform. Considering one has a multiplication of 2 functions in the laplace Domain
$G(s) F(s)= \exp(-a s) F(s)$
To transform that into the time domain one can simply transform $G(s)$ and $F(s)$ separately, which results in the convolution integral
\begin{align} \int_0^t \delta(t-a)f(t) \ dt &=\int_0^t \delta(t-\tau-a) f(\tau) \ d\tau \end{align}
I am a bit confused with the boundaries. I know the identity of
$$\int_{-\infty}^\infty \delta(t-a) f(t) \ dt = f(a)$$
In that case the boundaries are from $-\infty$ to $\infty$. Unfortunately in my case I integrate from $0$ to $t$, does that still results to the same solution of $f(a)$ or does that result in $f(t-a)$?