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Simpler intuitive opening, use pictures created by William Beason
'''Continuity''' is a concept central to [[calculus]], [[advanced calculus]] and [[topology]]. 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.
{{Template:Math-e}}
'''Continuity''' of functions is a concept central to [[calculus]], [[advanced calculus]] and [[topology]].
Put simply, a mathematical [[function]] is '''continuous''' if its graph can be drawn without lifting the pen from the paper. In the figures below, the graph on the left is a continuous function; the graph on the right is not.
[[Image:Br-cont-function.png]] [[Image:Br-discont-function.png]]
The function on the left is:
:<math>f(x) = x^3 - 3x^2 + 2x + 1\,</math>
The function on the right is:
:<math>f(x) = x^3 - 3x^2 + 2x + 1\,</math> for x <math>\le</math> 2
:<math>f(x) = x^3 - 3x^2 + 2x - 1\,</math> for x <math>>\,</math> 2
==More precise definition==
{{Template: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]], a [[function]] ''f(x)'' is said to be '''continuous''' at point ''c'' if ''f(c)'' equals the limit of ''f(x)'' as x approaches c from both the positive and negative directions.
In [[calculus]], a [[function]] ''f(x)'' is said to be '''continuous''' at point ''c'' if ''f(c)'' equals the limit of ''f(x)'' as x approaches c from both the positive and negative directions.