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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).
Continuous variations have a wide range of possibilities. For example, your height is a continuous variation. There are many options (for example you could be 5'9, 4'6, 6'1) rather than an either/or situation. Discrete variations have only two possibilities. They can be thought of as "either/or" situations. For example, you can either roll your tongue or you can't. There is no grey area or in-between.
discrete
discontinuous