I'm no mathematician, and my understanding of the subject is limited. How is the discrete Fourier transform unrelated to its continuous counterpart? Their definitions are extremely similar...

From a semantic point of view, it is nice that all these transform bear the same name, but historically, it is a bit unfair to attribute Fourier analysis to Fourier, who didn't discover it (even though he greatly developed the subject).

I know there's an eponymous law that describes this phenomenon, and it is of course not named after its inventor, but I can't remember its name.