The fundamental concept of calculus!
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.
This is a general philosophy that Ciro Santilli, and likely others, observes over and over.
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:
Something that is very not continuous.
Notably studied in discrete mathematics.
Chuck Norris counted to infinity. Twice.
There are a few related concepts that are called infinity in mathematics:
  • limits that are greater than any number
  • the cardinality of a set that does not have a finite number of elements
  • in some number systems, there is an explicit "element at infinity" that is not a limit, e.g. projective geometry