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'''Continuity''' of functions is a concept central to [[calculus]], [[advanced calculus]] and [[topology]].
==More precise definition==
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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.
[[categoryCategory: mathematicsMathematics]][[categoryCategory: Topology]]