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Between any two continuous data values there may be an infinite number of others. It sometimes makes sense to treat numeric data that is properly of one type as being of the other.
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Some very interesting further points were made in response to this follow-up question. Data on a variable are typically assumed to be drawn from a random variable.
"There may potentially be an infinite number of those values, but each is distinct and there's no grey area in between" -- it's actually perfectly possible to have a discrete distribution with distinct values, and yet at the same time, for any two distinct values you pick, always have more values between them ('grey area' in a sense). The random variable is continuous over a range if there is an infinite number of possible values that the variable can take between any two different points in the range.