Continuous function - Revision history

DavidB4-bot: clean up & uniformity

2016-07-13T06:52:22Z

clean up & uniformity

GregG: /* More precise definition */ remove duplicate topological definition, add examples of continuous but not differentiable functions

2015-08-19T00:22:14Z

More precise definition: remove duplicate topological definition, add examples of continuous but not differentiable functions

GregG: /* Metric Spaces */ to -> two

2015-08-18T23:19:25Z

Metric Spaces: to -> two

GregG: /* More precise definition */ add topological definition of continuous function

2015-08-18T23:19:05Z

More precise definition: add topological definition of continuous function

Aschlafly: /* More precise definition */Wikify

2015-08-18T21:14:02Z

More precise definition: Wikify

FrankC: /* Interesting examples */ another amusing example: a function which is continuous in one point only

2010-07-13T15:47:49Z

Interesting examples: another amusing example: a function which is continuous in one point only

FrankC: Interesting Example

2010-07-07T17:10:57Z

Interesting Example

FrankC: /* More precise definition */ ε-δ-definition for metric spaces

2010-07-07T17:01:18Z

More precise definition: ε-δ-definition for metric spaces

PatrickD: Simpler intuitive opening, use pictures created by William Beason

2009-11-02T03:51:52Z

Simpler intuitive opening, use pictures created by William Beason

Aschlafly: added an overall introduction that touches on calculus, advanced calculus, and topology; improvements welcome

2009-11-01T04:03:39Z

added an overall introduction that touches on calculus, advanced calculus, and topology; improvements welcome

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-{{Template:Math-e}}
+{{Math-e}}
 '''Continuity''' of functions is a concept central to [[calculus]], [[advanced calculus]] and [[topology]].
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 ==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.
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 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]]

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

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 === 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>

← Older revision		Revision as of 23:19, August 18, 2015
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	A continuous function maps a [[compact space]] to a [[compact space]].		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 ===		=== Metric Spaces ===

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 {{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, 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.

← Older revision		Revision as of 03:51, November 2, 2009
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−	'''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.

← Older revision		Revision as of 04:03, November 1, 2009
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−	~~{{merge\|~~calculus}}	+	'''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.
		+
	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.