The delta function has the fundamental property that
\begin{align} \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a) \end{align}
and, in fact, \begin{align} \int_{a-\epsilon}^{a+\epsilon}f(x)\delta(x-a)dx=f(a) \end{align} How change these formula if $a$ is not in the domain of integration? \begin{align} \int_{-\infty}^{a-\epsilon}f(x)\delta(x-a)dx + \int_{a+\epsilon}^{\infty}f(x)\delta(x-a)dx\end{align}
Also if $a$ is one of upper or lower bound?