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Continuous functions

Continuity is expressed in terms of neighborhoods: f is continuous at some point x ∈ X if and only if for any neighborhood V of f (x), there is a neighborhood U of x such that f (U) ⊆ V. Intuitively, continuity means no matter how "small" V becomes, there is always a U containing x that maps inside V and whose image under f contains f (x). This is equivalent to the condition that the preimages of the open (closed) sets in Y are open (closed) in X. In metric spaces, this definition is equivalent to the ε – δ-definition that is often used in analysis.

An extreme example: if a set X is given the discrete topology, all functions

to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology and the space T set is at least T, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.