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{{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==

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{{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.

An example of a discontinuous function is Y = 1/X, which has no value for X = 0; also the limits of the function as X approaches zero from each side are different.

An example of a discontinuous function is Y = 1/X, which has no value for X = 0; also the limits of the function as X approaches zero from each side are different.

A [[differentiable function]] is always continuous, but a continuous function is not always differentiable.

A [[differentiable function]] is always continuous, but a continuous function is not always differentiable. For example, the function <math>f(x) = |x|</math> is continuous everywhere but not differentiable at <math>x = 0</math>.  A more extreme example is the Weierstrass function, which is continuous everywhere but is differentiable only on a measure zero set.

 

A function f: X -> Y mapping elements in a [[topological space]] X to a topological space Y is continuous if for every [[open set]] U in Y, the inverse image of U under f is an open subset of X.

A continuous function maps a convergent [[sequence]], [[net]], or [[filter]] to a convergent sequence, net, or filter, respectively.

A continuous function maps a convergent [[sequence]], [[net]], or [[filter]] to a convergent sequence, net, or filter, respectively.

=== Metric Spaces ===

=== Metric Spaces ===

Let <math>X\,</math> and <math>Y\,</math> be to [[metric space]]s, and <math>f: X \rightarrow Y</math> a function between these two sets. Then <math>f\,</math> is ''continuous'' in <math>x_0 \in X</math> if for all <math>\epsilon > 0\,</math> there is a <math>\delta > 0\,</math> such that for all <math>x\,</math> with

Let <math>X\,</math> and <math>Y\,</math> be two [[metric space]]s, and <math>f: X \rightarrow Y</math> a function between these two sets. Then <math>f\,</math> is ''continuous'' in <math>x_0 \in X</math> if for all <math>\epsilon > 0\,</math> there is a <math>\delta > 0\,</math> such that for all <math>x\,</math> with

::<math> |x - x_0| < \delta\,</math>

::<math> |x - x_0| < \delta\,</math>

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]]