What are distributions really? If they are considered to be generalisation of function, as for examples real numbers are for rationals, then it must be possible too reinterpret any “classical” function $f(x)$ in terms of a distribution.
I don’t mean the identification $f \mapsto u_f(g):= {\displaystyle \int_{\mathbb R^n} f(x)g(x)dx}$
Addendum. Why? For two reasons:

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*It’s not always possible to define a distribution as an integration over a test function (e.g. Dirac delta)

*The purpose of generalising something is to add stuff without changing the previous stuff. For example you can generalise natural number with rationals, and see a natural number as a rational with $1$ as denominator.

In the same way, a classical function that eats numbers $f(x)$ should be seen, for instance, as a particular case of functional $f(g)$ that eats the identity function $g(x)=x$ or something like that. I don’t know if it’s clear what I’m saying.
 A: Perhaps I have an inkling of what is desired...
By this point, my viewpoint on pointwise evaluation of functions (when possible and when meaningful...!!!) is that it is an example of a linear functional on various vector spaces of functions. For example, for continuous functions $f$ (say on $\mathbb R^n$), for fixed $x_o\in\mathbb R^m$, the map $f\to f(x_o)$ is a linear map from continuous functions to scalars (real or complex).
In particular, although for many purposes "functions" accept inputs $x_o\in\mathbb R^n$ and produce real or complex outputs, this is maybe not the universally useful way to think about "functions".
Already looking at $L^2$ "functions", pointwise values are simply not well-defined, because these functions are really equivalence classes of pointwise-valued functions, where being equal off a set of measure $0$ means that two functions are "the same". Pointwise evaluation at a single point simply has no meaning on these equivalence classes. But that's ok! It turns out that we really don't have reasons to evaluate $L^2$ functions at points, but, rather, mostly to integrate them against each other. No disaster.
And, further, pointwise evaluation $f\to e_{x_o}f=f(x_o)$ of continuous functions is a continuous linear functional! Not just an algebraic functional. This is good, because limits are respected: $e_{x_o}(\lim_n f_n)=\lim_n e_{x_o}(f_n)$, where the left-hand-side limit is uniformly-on-compacts, and the right-hand-side limit is in scalars.
But/and if we try to define pointwise evaluation on $L^2$, using the fact that continuous $L^2$ functions are dense in $L^2$, we will find that (in general) $e_{x_o}(\lim_n f)\not=\lim_n e_{x_o}(f_n)$, when the left-hand-side limit uses $L^2$ limits of continuous functions. :)
But/and we might desire to present some of these functionals as "integrals" or generalizations or limits thereof. While there is no pointwise-valued function $\delta$ such that $\int \delta(x)\,f(x)\;dx=f(0)$ for continuous $f$, but there are various sequences $f_n$ of continuous functions such that $\lim_n \int f_n(x)\,f(x)\;dx=f(0)$. Thus, we can imagine that the limit might be moved inside the integral, and that $\lim_n f_n$ could make sense as some sort of generalized function... even though that limit cannot be pointwise-uniform. (As an exercise, we can arrange that the pointwise limit does exist everywhere, and is $0$, but the limit of the integrals is still $f(0)$, which need not be $0$.)
So it may be that changing perspective, to think of "functions" (of various sorts) not as so much as evaluatable at points, but as forming topological vector spaces admitting various sorts of continuous linear functionals. Pointwise eval'n makes sense for certain types of "functions", but not for others.
(And maybe integration-against, especially abstracted, is operationally more important/useful than pointwise-evaluation...)
When I teach real analysis or functional analysis, I do try to make the point that "pointwise-valued functions" (as traditional as they are) are not the ultimate way to think about functions... especially if/when we allow weak limits, etc. So, yes, kinda too bad that the set-theory pseudo-final description of a function $f:X\to Y$ as its graph, a subset of the Cartesian product $X\times Y$ meeting certain conditions, is not really sufficiently inclusive. :)
