Bytes--Measured or Counted
A file size is measured in bits, you cannot have a portion of a bit, only a whole bit, it is a discrete quantity. Something is continuous if it can can have any possible value in a range, bounded or not. Something can be discrete and at the same time infinite.

The best way to think about discrete or continuous is if for any two values you can find a value between them then it is continuous, otherwise it is discrete.