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Line 1: − A function f: X -> Y mapping elements in the [[topological space]] X to the topological space Y is continuous if for every [[open set]] in Y, the inverse image of Y under f is an open subset of X.+ − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
clean up & uniformity
{{Math-e}}
[[category: mathematics]]
'''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==
{{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.
Another way of understanding this is by recognizing that a discontinuous function over a specific interval is one that has a gap in the interval, or one having different limits at a particular point depending on whether it is approached from the positive or negative directions.
A simple example of a continuous function would be Y = 2X + 5.
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. 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 continuous function maps a convergent [[sequence]], [[net]], or [[filter]] to a convergent sequence, net, or filter, respectively.
A continuous function maps a [[compact space]] to a [[compact space]].
=== Topological definition ===
Let <math>X</math> and <math>Y</math> be [[topological space]]s. A function <math>f\colon X \to Y</math> is continuous if for every set <math>U</math> that is open in <math>Y</math>, the preimage <math>f^{-1}(U)</math> is an open set in <math>X</math>.
=== Metric Spaces ===
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>
we have
::<math>|f(x) - f(x_0)| < \epsilon \,</math>.
This is the notorious <math>\epsilon-\delta-\,</math>''definition'' of continuity. Especially, it works for the metric spaces <math>\mathbb{R}\,</math> and <math>\mathbb{R}^n\,</math>, and it is used in any college level course on calculus.
== Interesting examples ==
The [[characteristic function]] of <math>\mathbb{Q}\,</math> in <math>\mathbb{R}\,</math>, <math>\chi_{\mathbb{Q}} : \mathbb{R} \rightarrow \mathbb{R}</math>, defined as
<math>\chi_{\mathbb{Q}}(x)</math><math> = \left\{\begin{matrix}
\,1 \quad &: &\quad x \in \mathbb{Q} \\
\,0 \quad &: &\quad x \in \mathbb{R} \setminus \mathbb{Q}
\end{matrix}\right.</math>
is nowhere continuous.
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:Topology]]