Let $\delta$ be the Dirac Delta distribution and $f$ be Riemann-integrable. Then it is said $(f\*\delta)(t) = f(t).$

But, I'm wondering if this can be proven. The dirac delta distribution converges to an object that's $0$ everywhere except at $y = 0.$

So this leave us with a question, what is the value of the integral at 0? We evaluate $\lim\_{R \rightarrow 0} \int\_{-R}^{R}f(t,y)\delta(y)dy$ which is

$$ \lim\_{R \rightarrow 0} \lim\_{n \rightarrow \infty} \sum\_{i=1}^{n}f(t,y\_i)\delta(y\_i)\Delta y,$$ but I'm unsure what to do from here or where to find a reference.