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{{Template:Math-e}}

{{Math-e}}

'''Continuity''' of functions is a concept central to [[calculus]], [[advanced calculus]] and [[topology]].

'''Continuity''' of functions is a concept central to [[calculus]], [[advanced calculus]] and [[topology]].

==More precise definition==

==More precise definition==

{{Template:Math-h}}

{{Math-h}}

In calculus, continuity is defined based on limits.  In advanced calculus, continuity is defined using neighborhoods or sequences.  In [[topology]], a function is continuous if the inverse image of every open set in the function's range is also an open set in the function's domain.  In all three fields of [[mathematics]], the unifying characteristic of continuity is that points near each other in a set or domain are mapped by the continuous function to points that are near each other in the corresponding set or range.

In calculus, continuity is defined based on limits.  In advanced calculus, continuity is defined using neighborhoods or sequences.  In [[topology]], a function is continuous if the inverse image of every open set in the function's range is also an open set in the function's domain.  In all three fields of [[mathematics]], the unifying characteristic of continuity is that points near each other in a set or domain are mapped by the continuous function to points that are near each other in the corresponding set or range.

Similarly, the function <math>f(x) = x \cdot \chi_{\mathbb{Q}}(x)</math> is continuous only in 0, and discontinuous everywhere else.

Similarly, the function <math>f(x) = x \cdot \chi_{\mathbb{Q}}(x)</math> is continuous only in 0, and discontinuous everywhere else.

[[category: mathematics]]

[[Category:Mathematics]]

[[category: Topology]]

[[Category:Topology]]