continuous

Naively, continuity is a property of collections of points, or of numerical values, corresponding to a lack of gaps or jumps. For instance, a line would be called continuous if it could be drawn without ever lifting one's pencil from the paper. A set of values would be called continuous if it included every possible value.

These naïve notions of continuity are extended in various ways in different fields of mathematics. Most commonly, the set of real numbers is used as the model for those variable phenomena, such as time or distance, that are taken to be continuous. This is because the real numbers are geometrically complete, meaning that there is a one-to-one correspondence between the set of real numbers and the set of conceivable physical measures or magnitudes. To some extent, every field of mathematics is characterized by the ways in which it abstracts the notion of continuity to its needs.

Analysis

A function is continuous if it maps ‘close’ values in the domain to ‘close’ values in the range. Formally, \(f\) is continuous at a point \(x_0\) of its domain if for any given positive \(\varepsilon\) we may find a \(\delta\) so that whenever \(x\) is within a \(\delta\) neighborhood of \(x_0\) then \(f(x)\) is within an \(\varepsilon\) neighborhood of \(f(x_0)\).

Statistics

A random variable is said to be continous if between any two distinct values of the variable a measurement may return a third value that lies strictly between them.

Topology

A transformation of one topological space into another is continuous if the inverse image of every open set is open, or equivalently if the inverse image of every closed set is closed.