The reason why the epsilon delta definition is so venerated is that it fits directly into well known methods of the formalization of mathematics, making the notion completely precise.
Basically, continuity, or higher order conditions like differentiability seem to impose greater constraints on problems, which make them more solvable.
Some good examples of that:
- complex discrete problems:
- simple continuous problems:
- characterization of Lie groups
Something that is very not continuous.
Notably studied in discrete mathematics.
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