Okay, here are the ingredients to this question.
Me: 60 years old. 39 years ago I took two semesters of Real Analysis using the Royden textbook. Rusty is an understatement. But I am still quite anal and OCD. I am also an electrical engineer, works in signal processing. DSP and Linear System Theory are important to me. I have also had two semesters (as a grad student) of Functional Analysis (using the Kreyszig text) and multiple courses in probability, random variables, and random processes (a.k.a. "stochastic processes").
Electrical Engineers (and I suspect many physicists) essentially treat $\delta(x)$ as a "function". But it isn't. One thing I remember from R.A. is that if
$$ f(x) = g(x) $$
almost everywhere in $E$, then
$$ \int_E f(x) \, dx \ = \ \int_E g(x) \, dx $$
problem is, of course, that electrical engineers (and their professors) like to think of
$$\begin{align} f(x) & = \delta(x) \\ g(x) & = 0 \end{align} $$
and that
$$ \int\limits_{-1}^{+1} f(x) \, dx = 1 \quad \ne \quad \int\limits_{-1}^{+1} g(x) \, dx = 0 $$
yet $f(x) = g(x)$ everywhere except at one single value of $x$ .
Now I have heard (or read) that the "Dirac delta function is not really a function but is a 'distribution' or a 'functional'." And I understand the meaning of the terms "distribution" in the context of random variables and "functional" in the context of metric spaces, normed spaces, etc. Is that the given usage of these two terms regarding $\delta(x)$?
Question 1: Is the usage of the notation
$$ \int\limits_{-\infty}^{+\infty} f(x) \, \delta(x) \, dx $$
a misnomer? There is no integration going on. It's just a linear functional that maps the function $f(x)$ to the number $f(0)$. Now EEs and maybe physicists will comfortably look at that as an integral that is the same as
$$ \int\limits_{-\infty}^{+\infty} f(0) \, \delta(x) \, dx = f(0)\int\limits_{-\infty}^{+\infty} \delta(x) \, dx = f(0) $$
But since $\delta(x)$ is not a function at all, what do mathematicians mean with that notation?
Question 2: How fatal is it for electrical engineers and physicists to consistently treat the Dirac delta function simply as a limit of "nascent deltas" such as
$$ \delta(x) \ \triangleq \ \lim_{\sigma \to 0^+} \frac{1}{ \sigma} \operatorname{rect} \left( \frac{x}{\sigma} \right) $$
where $ \operatorname{rect}(x) \triangleq \begin{cases} 1 \quad |x|<\frac{1}{2} \\ \frac{1}{2} \quad |x|=\frac{1}{2} \\ 0 \quad |x|>\frac{1}{2} \\ \end{cases} $
or
$$ \delta(x) \ \triangleq \ \lim_{\sigma \to 0^+} \frac{1}{\sigma} \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x}{\sigma}\right)^2} $$
What is gonna kill us to simple-mindedly treat the Dirac delta as such a function? A function that is zero almost everywhere, yet it integrates to be equal to 1 (where the function that is zero everywhere integrates to be 0).
If we do that, within our own disciplines, what mathematical problem might crop up that kills us?
This is not exactly the same but smells a lot like this concern from Richard Hamming:
“Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.”
I might ask the same question regarding the mathematician's and the engineer's understanding of the Dirac delta function. How might a mathematician answer that question?