I'm having trouble with the different infinities involved in the Discrete and Continuous Fourier Transforms.
In the DFT, we have a finite number $N$ time domain samples $x(i), 0\leq i<N$, which transform to the same finite number of frequency coefficients $X(k), 0\leq k<N$. This is perfectly acceptable: there are the same number of degrees of freedom in each domain.
In the continuous FT, we have a continuous function of infinite extent in the time domain: $x(t), -\infty < t < \infty$ and a continuous function of infinite extent in the frequency domain, $X(\omega), -\infty < \omega < \infty$. Again, the functions in the two domains are the same type of animal, with the same (albeit infinite) number of degrees of freedom.
Where things get puzzling is the mixed cases, for example the Discrete Time Fourier Transform (DTFT). Here we have an infinite sequence of discrete samples in the time domain: $x(i), -\infty <i < \infty$, and a periodic function of continuous frequency in the frequency domain, $X(\Omega), 0\leq \Omega < 2 \pi $. I could have chosen any other equivalent range for $\Omega$; the point is that it is free to take any form over a given $2\pi$ interval but is constrained to repeat itself every $2\pi$.
I used to be comfortable with this: to an engineer, it seems plausible that an infinite discrete list might have the same number of degrees of freedom as a continuous function over a finite interval. But now I know that the number of different time domain samples is only countably infinite, since they are indexed using the integers, whereas it turns out that even an finite interval of the real numbers contains an uncountably infinite number of elements. So the number of frequency domain coefficients is uncountably infinite.
It seems therefore that the frequency domain representation has more degrees of freedom than the time domain. Is this correct, and if so how can it be reconciled? Are there an infinite number of frequency domain functions that somehow don't map to a time domain sequence? What happens if one tries to transform them?
A colleague argued that the DTFT is simply a limiting case of the DFT; that as the time domain sequence gets longer, the frequency domain resolution gets higher and higher, until you have the infinitely long sequence and the continuous function. This suggests a more fundamental formulation of my problem: Let $S_N$ be the set of numbers $(i/N), 0\leq i<N$, and let $N\to \infty$. In the limit, there are countably infinite numbers in the set any yet they appear to map to a finite continuous interval. But this apparent paradox was dismissed by a mathematician colleague on the basis that the limit does not exist, or has no meaning.
The space $L^{2}[0,2\pi]$ has a countable complete orthonormal basis. So does $L^{2}(\mathbb{R})$. So the dimensions of these spaces is same, at least in terms of Hilbert spaces dimension. Equivalently, both are separable, meaning that they have countable dense subsets. In either case, you can consider the set of all step functions $$ S_{a,q,k}=\sum_{n=1}^{k}a_{n}\chi_{[q_{n},q_{n+1})} $$ where $a_{n}$, $q_{n}$ are rational numbers. Such functions are dense in either $L^{2}$ space, and there are a countable number of such functions.
There are several ways in which one may view the Fourier transform as a limit of the discrete Fourier transform, along with the inverse formula. It's a bit tedious to make this all rigorous, but it was done rather generally and quite rigorously by E. C. Titchmarsh, a brilliant student of G. H. Hardy and a Professor at Oxford University. Titchmarsh studied general sorts of Sturm-Liouville problems in a classical, pointwise way, and he was one of the first to rigorously show how to rigorously take the limit of Fourier expansions on finite intervals in order to study such problems on infinite intervals. You can find this in the classic work Eigenfunction Expansions Associated with Second Order Differential Equations published in 1946 by Oxford University Press.
The first colleague you mention laid out a technique that is along the same line as that used by Titchmarsh.