HW12 DFT/Continuous Fourier Transform Relationship

The Question to answer here is: What is the relationship between the continuous Fourier Transform, and the discrete one? For those of you who recognize the squiggly codes of mathematical formulas, observe:

Here is the continuous Fourier Transform
$X(f)=\int _{-\infty }^{\infty }x(t)e^{-j2\pi ft}\,dt$
Here is the Discrete Fourier Transform
$X(f)=\sum _{n=0}^{N-1}x(n)e^{\frac {-j2\pi nm}{N}}$

In words, you may describe the Discrete Fourier Transform (DFT) as a sampled Continuous Fourier Transform. The Discrete represents a finite amount of points on the continuous transform - enough so that you can reconstruct the signal to the correct amount of accuracy that you need, and prevent aliasing (see HW #11)

So why would you make a Discrete Fourier Transform? Why not just keep all the information you had in the first place with the continuous? Well, you could, you'd just need an infinite amount of memory on your computer to store the infinite amount of points on the continuous waveform. And that would leave no room for your precious Mp3's and movie files. You wouldn't want that, would you?

The Discrete Transform allows us to chop that bit ol' continuous ones into a bunch of ones and zeros that will fit on your computer along with your Vanilla Ice and Mariah Carey and Michael Jackson Anthologies.

But, if you're like me, this probably isn't enough description or motivation to truly understand the relationship between DFT's and CFT's. You want pictures, loud and clear. Here they are: