Suppose $f:\mathbb{R}\rightarrow\mathbb{C}$. We'd like to take the Fourier Transform of this function, defined as $$F(s) = \int_{-\infty}^\infty f(x)e^{-i \, 2\pi \, s \, x}dx.$$ However, a computer can't do this in general. Instead, we "sample" the function and then compute the Discrete Fourier Transform. Sampling is modeled as multiplication with a comb function $\text{III}(x)=\sum_{n=-\infty}^\infty\delta(x-n)$.
So we take the Fourier Transform of $f\,\text{III}$.
At this point I'm confused about a few things.
1) Shouldn't we model sampling as multiplication with a comb and a rect function like this: $f\,\text{III}\,\Pi_\Delta$ where $$\Pi_\Delta(x)=\left\{\begin{array}{ll}1 & \text{ if } |x|/\Delta < 0.5\\0 & \text{otherwise}\end{array}\right..$$ Don't we need this in order to account for the fact that we have a finite number of samples?
2) How does the Fourier Transform of $f\,\text{III}$ relate to the Discrete Fourier Transform of the samples?
Thank you.