Maybe it would be a good idea to point out the following more clearly:
- A rigorous treatment of delta(t) is beyond the scope of this course material. (Same with most EE textbooks.)
- Fortunately, for our purpose it's enough to "know the rules".
Or to quote the man himself:
Thus delta(x) is not a quantity which can be generally used in mathematical analysis like an ordinary function, but its use must be confined to certain simple types of expression for which it is obvious that no inconsistency can arise.
(P.A.M. Dirac, The Principles of Quantum Mechanics, 4th Ed., §15, p.58)
If someone wants to dig deeper, A. Vretblad - "Fourier Analysis and Its Applications" (Ch.8) has IMHO a quite accessible introduction to distribution theory.