Another branch of math that describes properties that change only in increments is...accounting.https://twitter.com/St_Rev/status/783533771199045632 …
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Not sure I understand the question. One usually encounters continuity first in calculus, but...
...while continuity is *important*, it's more of a niceness criterion. A mature treatment of integration uses measure theory.
Measure spaces and topological spaces are different things, though of course you'd prefer your space to be both!
nb: I'm an algebraist and am pretty bad at analysis.
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