Question

are percentages continuous or discrete

Answer 1

Discrete.

Answer 2

I think it depends on whether they are of countable entities, thus discrete or continuous ones (like water), thus continuous.

Answer 3

continuous means we can always find a number between any other two numbers. discrete means we cannot.
given any two percentages, can we always find a percentage that is between those two given values?

Answer 4

"find" is tricky. If you are counting, you will not find percentages corresponding to non-integer counts. Percentages seem capable of being any value, before you describe the experiment.