What is the minimal requirement for a Fourier transformation?

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I have a lot of comprehension questions that I can't really figure out by googling:

  1. The difference between a Fourier transformation and a Fourier series. Am I correct in thinking that the difference: transforms are for continous cases and series are for discrete cases ?

  2. What functions can be Fourier transformed? Which not? I read/learned once that piecewise continuity is enough for a function to be transformed as long as it can be extended to 2 - pi periodic continuity (so the pieces can be "stringed together to a periodic wave")

Is this correct?