The Dirac delta function, despite being a somewhat abstract and non-rigorous entity in its initial presentation, has been rigorously defined within the framework of distribution theory. Over the years, it has proven invaluable in various branches of physics and engineering due to its ability to elegantly handle problems that would be much more complicated without it.

It’s initially looks so simple but then mind-bending a little bit that I rely on substitution to figure out:

Start from simple example

And we find delta function is symmetrical