What should I do if limits of integration are ones where Dirac Delta gets Infinite? I am working on an integral involving Dirac delta function. I am from Physics background. I know that if I integrate from a to b and in between there is a point where argument of Dirac delta function is zero then the answer to this integral is unity. What if at say 'a' the argument of Dirac Delta Function is zero. What should I do in that case? What if I have a Riemann Integrable Function along with Dirac delta function? Please help.
Consider a graph of the delta function (distribution) to obtain:

$$\int_a^b\delta(x-t)\,\mathrm dx 
= \cases{0&amp;$t&lt;a,\text{ or }t&gt;b$\\
         1&amp;$a&lt;t&lt;b$\\
\color{red}{1/2}&amp;$\color{red}{t=a},\text{ or }t=b$}$$

So that
$$\int_a^bf(x)\delta(x-a)\,\mathrm dx 
= \tfrac12f(a)$$
Consider a graph of the delta function (distribution) to obtain:

$$\int_a^b\delta(x-t)\,\mathrm dx 
= \cases{0&amp;$t&lt;a,\text{ or }t&gt;b$\\
         1&amp;$a&lt;t&lt;b$\\
\color{red}{1/2}&amp;$\color{red}{t=a},\text{ or }t=b$}$$

So that
$$\int_a^bf(x)\delta(x-a)\,\mathrm dx 
= \tfrac12f(a)$$