continuous function

A continuous function is one that is continuous everywhere it lives — at every single point of its domain. Where pointwise continuity is a local promise (this one point is well-behaved), being a continuous function is the global statement that the promise holds throughout.

Formally, f: D -> R is a continuous function when it is continuous at a for every a in D. Equivalently, in topological language, the preimage of every open set is open (relative to the domain). This open-set formulation is what generalizes continuity far beyond the real line, to metric and topological spaces, where there may be no epsilon or delta at all.

Continuous functions are closed under the everyday operations: sums, differences, products, quotients (where the denominator is nonzero), and compositions of continuous functions are continuous. A common trap is the word ‘continuous’ applied to a function with a restricted domain — for example 1/x is a perfectly continuous function on its domain (all x not equal to 0), even though it ‘blows up’ near 0, because 0 is simply not a point of the domain.