Is the solution of the following integral $0$ or $1$ or something else? And why so?
$$ \int_{0}^{t} f(\tau) \;\delta(\tau) \;d\tau$$
with
$\;\;\;t \in \mathbb{R}\,,\; f: \mathbb{R}\rightarrow\mathbb{R}\;\;\;f$ continuous
$\;\;\;\delta(x)=\left\{\begin{matrix}+\infty\;\;for\,\,x=0\\0\;\;\;\;\;\, for\,\,x\neq0\end{matrix}\right.$