quality:corruption, information:highlighting
Filters and prefilters

Continuity can also be characterized in terms of filters. A function f: X → Y {\displaystyle f:X\to Y} is continuous if and only if whenever a filter B {\displaystyle {\mathcal {B}}} on X {\displaystyle X} converges in X {\displaystyle X} to a point x ∈ X, {\displaystyle x\in X,} then the prefilter f (B) {\displaystyle f({\mathcal {B}})} converges in Y {\displaystyle Y} to f (x). {\displaystyle f(x).} This characterization remains true if the word "filter" is replaced by "prefilter."