Conditions and techniques for discretizing a real function Sampling theorem states:

If a function $x(y)$ contains no frequencies higher than $B$ hertz, it is completely determined by uniform samples taken at less than $1/2B$ units apart.

Original sampling theorem has $x$ has a function of time $t$, but mathematically there is no requirement for it to be a function of time in order to apply sampling theorem.
Sampling theorem allows us to sample a signal and then reconstruct it without any errors if the above conditions are satisfied.
The sampled signal would look like $x[n]$. Here $n$ is discrete, but $x$ can be real. So sampling can be thought of mapping a real-valued function with real inputs, $A$, to a real-valued function with discrete inputs, $B$, such that I can get back the original signal $A$ from $B$ using interpolation techniques, subject to certain conditions (the maximum frequency condition).
We are taught in college that after sampling we do quantization, which makes $x[n]$ take on discrete values, and this introduces quantization errors. The final aim is to discretize $x[n]$ in both amplitude and time, so we can process it easily on a digital computer. While discretizing in time can be done such that function can be reconstructed without any errors, subject to maximum frequency condition, the same has not been said for discretizing in amplitude.
I do understand that Analog-to-Digital Converters (ADC) physically limits the accuracy of the quantization process, and results in quantization errors. This, however, does not stop one from converting real valued functions to integer functions in the field of mathematics. So for this question, the physical limits of ADCs can be ignored as I am only interested in the mathematics.
My question is:
Can I have a function $x(y)$ in which both $x$ and $y$ are continuous, from which I can get a mapping to a function $a(b)$, where $a$ and $b$ are both discrete, such that I can reconstruct the original function, $x(y)$, using interpolation techniques? If so, what are the required conditions? (Pointing to a relevant resource and mentioning a little context/background information is sufficient.)
I am not restricting the question to sampling theorem, other techniques are also welcome.
What I have tried:
https://dsp.stackexchange.com/a/76143/47984
Asked a related question, namely, can sampling theorem be applied to quantization, on DSP Stack Exchange.
From a comment by a user, if the function is bijective, then this can be done.
While that comment seems fine intuitively, it lacks mathematical justification. Also that post does not address this: Does applying sampling theorem in time domain allow one to apply it again for the amplitude domain? (This question is precisely the question I ask here.)
One DSP.SE user has mentioned in his answer:

So I think the constraint required for both of your reconstruction theorems to hold only includes infinite DC signals of an exact quantization integer multiple in value. e.g. a constant.

This is easy to understand, a real function $f(x) = k$, where $x$ is real and $k$ is a constant. This real function can be expressed as a discrete function $f[n] = k_1$, where $k_1$ and $n$ are both integers. From $f[n]$, $f(x)$ can be easily got back: multiply $k_1$ by a constant $m = k/k_1$, and replace $n$ with real valued $x$. The question is: Is this the only kind of function for which this works?
 A: *

*The theorem of Shannon says that if a function $f(t)$ have Fourier transform $F(w)=\int^{+\infty}_{-\infty}f(t)e^{-i w t}dt$ such that $F(w)=0$, for all $|w|\geq w_c$ (Band Limited), then if $f_n=f\left(\frac{n\pi}{w_c}\right)$, we have
$$
f(t)=\sum^{+\infty}_{n=-\infty}f_n\frac{\sin(w_ct-n\pi)}{w_ct-n\pi}\textrm{, }\forall t\in\textbf{R}\textrm{. (This is Exactly.)}\tag 1
$$
Since now $F(w)=\int^{w_c}_{-w_c}f(t)e^{-it w}dt$, this function is analytic in all $\textbf{R}$ (it possess any high order derivatives in $\textbf{R}$).


*The theorem also have oposite interpretation
If a function $f(t)$ is such $f(t)=0$, for all $|t|>T$, then its Fourier transform $F(w)$ have development
$$
F(w)=\sum^{+\infty}_{n=-\infty}F\left(\frac{n\pi}{T}\right)\frac{\sin(wT-n\pi)}{wT-n\pi}\textrm{. (Exactly)}\tag 2
$$
But if $f(t)$ is zero in $|t|>T$, then it is not Band Limited and may have not Shannon development. For example $f(t)=p_T(t)=1$, if $t\in[-T,T]$ and $f(t)=0$, if $|t|>T$, then $F(w)=\frac{2\sin(wT)}{w}$ (and this is not equals zero in a region $|w|>w_c$). Hence $f(t)=p_T(t)=X_{[-T,T]}(t)$, can not represented with the series (1). Hence not all functions can be represented with the series (1).
Now consider an arbitrary function $f(t)$ and $a$ be a constant if we form the function
$$
g(t)=\sum^{+\infty}_{n=-\infty}f\left(\frac{n\pi}{a}\right)\frac{\sin(at-n\pi)}{at-n\pi}.
$$
Then $g(n\pi/a)=f(n\pi/a)$ and the spectrum of $g$ is $0$ for all $|w|>a$. Hence $g(t)$ is a band limited iterpolation of $f(t)$, but it is not $f(t)=g(t)$, for all $t$.
For other constructions see A.Melas, N.Bagis.
For the Paley-Wiener theorem see:
[Robert M. Young]. An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.
