Examples of continuous variation are anything that can be measured such as, shoe size, height, weight, hand span and diameter of limpit shells.
Discontinuous variation however is when there is a clear cut difference such as different colours or different species.
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It is an example of continuous variations.
How about f(x) = floor(x)? (On, say, [0,1].) It's monotone and therefore of bounded variation, but is not Lipschitz continuous (or even continuous).
discrete
hair colour
discontinuous