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In [[mathematics]], the notion of '''continuity''' of a function relates to the idea that the "value" of the function should not jump abruptly for any vanishingly "small" variation to its argument. Another way to think about a continuity of a function is that any "small" change in the argument of the function can only effect a correspondingly "small" change in the value of the function.
In [[mathematics]], the notion of '''continuity''' of a function relates to the idea that the "value" of the function should not jump abruptly for any vanishingly "small" variation to its argument. Another way to think about a continuity of a function is that any "small" change in the argument of the function can only effect a correspondingly "small" change in the value of the function.


==Formal definition of continuity==
==Formal definitions of continuity==
We can develop the definition of continuity from the <math>\delta-\epsilon</math> formalism which are usually taught in first year calculus courses to general topological spaces.
 
===Function of a real variable===
The <math>\delta-\epsilon</math> formalism defines limits and continuity for functions which map the set of real numbers to itself. To compare, we recall that at this level a function is said to be continuous at <math>x_0\in\mathbb{R}</math> if (it is defined in a neighborhood of <math>x_0</math> and) for any <math>\varepsilon>0</math> there exist <math>\delta>0</math> such that
:<math> |x-x_0| < \delta \implies |f(x)-f(x_0)| < \varepsilon. \,</math>
Simply stated, the [[limit of a function|limit]]
:<math>\lim_{x\to x_0} f(x)  = f(x_0).</math>
 
This definition of continuity extends directly to functions of a [[complex number|complex]] variable.
 
===Function on a metric space===
A function ''f'' from a [[metric space]] <math>(X,d)</math> to another metric space <math>(Y,e)</math> is ''continuous'' at a point <math>x_0 \in X</math> if for all <math>\varepsilon > 0</math> there exists <math>\delta > 0</math> such that
 
:<math> d(x,x_0) < \delta \implies e(f(x),f(x_0)) < \varepsilon . \,</math>
 
If we let <math>B_d(x,r)</math> denote the [[open ball]] of radius ''r'' round ''x'' in ''X'', and similarly <math>B_e(y,r)</math> denote the [[open ball]] of radius ''r'' round ''y'' in ''Y'', we can express this condition in terms of the pull-back <math>f^{\dashv}</math>
 
:<math>f^{\dashv}[B_e(f(x),\varepsilon)] \supseteq B_d(x,\delta) . \, </math>
 
===Function on a topological space===
A function f from a [[topological space]] <math>(X,O_X)</math> to another topological space <math>(Y,O_Y)</math>, usually written as <math>f:(X,O_X) \rightarrow (Y,O_Y)</math>, is said to be '''continuous''' at the point <math>x \in X</math> if for every [[open set]] <math>U_y \in O_Y</math> containing the point ''y=f(x)'', there exists an open set <math>U_x \in O_X</math> containing ''x'' such that <math>f(U_x) \subset U_y</math>. Here <math>f(U_x)=\{f(x') \in Y \mid x' \in U_x\}</math>. In a variation of this definition, instead of being open sets, <math>U_x</math> and <math>U_y</math> can be taken to be, respectively, a [[topological space#Some topological notions|neighbourhood]] of ''x'' and a neighbourhood of <math>y=f(x)</math>.
A function f from a [[topological space]] <math>(X,O_X)</math> to another topological space <math>(Y,O_Y)</math>, usually written as <math>f:(X,O_X) \rightarrow (Y,O_Y)</math>, is said to be '''continuous''' at the point <math>x \in X</math> if for every [[open set]] <math>U_y \in O_Y</math> containing the point ''y=f(x)'', there exists an open set <math>U_x \in O_X</math> containing ''x'' such that <math>f(U_x) \subset U_y</math>. Here <math>f(U_x)=\{f(x') \in Y \mid x' \in U_x\}</math>. In a variation of this definition, instead of being open sets, <math>U_x</math> and <math>U_y</math> can be taken to be, respectively, a [[topological space#Some topological notions|neighbourhood]] of ''x'' and a neighbourhood of <math>y=f(x)</math>.


This definition is a generalization of the <math>\delta-\epsilon</math> formalism which are usually taught in first year calculus courses to, among other things, define limits and continuity for functions which map the set of real numbers to itself. To compare, we recall that at this level a function is said to be continuous at <math>x_0\in\mathbb{R}</math> if (it is defined in a neighborhood of <math>x_0</math> and) for any <math>\varepsilon>0</math> there exist <math>\delta>0</math> such that
 
:<math> |x-x_0|\le \delta \implies |f(x)-f(x_0)| \le \varepsilon.</math>
Simply stated,
:<math>\lim_{x\to x_0} f(x)  = f(x_0).</math>


==Continuous function==
==Continuous function==
If the function ''f'' is continuous at every point <math>x \in X</math> then it is said to be a '''continuous function'''. An important ''equivalent'' definition, but perhaps less convenient to work with directly, is that a function <math>f:(X,O_X) \rightarrow (Y,O_Y)</math> is continuous if for any open set <math>U \in O_Y</math> (respectively, [[closed set|closed subset]] of ''Y'' ) the set <math>f^{-1}(U)=\{ x \in X \mid f(x) \in U\}</math> is an open set in <math>O_x</math> (respectively, a closed subset of ''X'').
If the function ''f'' is continuous at every point <math>x \in X</math> then it is said to be a '''continuous function'''. There is another important ''equivalent'' definition that does not deal with individual points but uses a 'global' approach. It may be convenient for topological considerations, but perhaps less so in classical analysis.  A function <math>f:(X,O_X) \rightarrow (Y,O_Y)</math> is said to be continuous if for any open set <math>U \in O_Y</math> (respectively, [[closed set|closed subset]] of ''Y'' ) the set <math>f^{-1}(U)=\{ x \in X \mid f(x) \in U\}</math> is an open set in <math>O_x</math> (respectively, a closed subset of ''X'').[[Category:Suggestion Bot Tag]]

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In mathematics, the notion of continuity of a function relates to the idea that the "value" of the function should not jump abruptly for any vanishingly "small" variation to its argument. Another way to think about a continuity of a function is that any "small" change in the argument of the function can only effect a correspondingly "small" change in the value of the function.

Formal definitions of continuity

We can develop the definition of continuity from the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta-\epsilon} formalism which are usually taught in first year calculus courses to general topological spaces.

Function of a real variable

The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta-\epsilon} formalism defines limits and continuity for functions which map the set of real numbers to itself. To compare, we recall that at this level a function is said to be continuous at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0\in\mathbb{R}} if (it is defined in a neighborhood of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} and) for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon>0} there exist Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta>0} such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x-x_0| < \delta \implies |f(x)-f(x_0)| < \varepsilon. \,}

Simply stated, the limit

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to x_0} f(x) = f(x_0).}

This definition of continuity extends directly to functions of a complex variable.

Function on a metric space

A function f from a metric space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,d)} to another metric space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Y,e)} is continuous at a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0 \in X} if for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0} there exists Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta > 0} such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,x_0) < \delta \implies e(f(x),f(x_0)) < \varepsilon . \,}

If we let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_d(x,r)} denote the open ball of radius r round x in X, and similarly Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_e(y,r)} denote the open ball of radius r round y in Y, we can express this condition in terms of the pull-back Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{\dashv}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{\dashv}[B_e(f(x),\varepsilon)] \supseteq B_d(x,\delta) . \, }

Function on a topological space

A function f from a topological space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,O_X)} to another topological space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Y,O_Y)} , usually written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:(X,O_X) \rightarrow (Y,O_Y)} , is said to be continuous at the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} if for every open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_y \in O_Y} containing the point y=f(x), there exists an open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_x \in O_X} containing x such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(U_x) \subset U_y} . Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(U_x)=\{f(x') \in Y \mid x' \in U_x\}} . In a variation of this definition, instead of being open sets, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_y} can be taken to be, respectively, a neighbourhood of x and a neighbourhood of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=f(x)} .


Continuous function

If the function f is continuous at every point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} then it is said to be a continuous function. There is another important equivalent definition that does not deal with individual points but uses a 'global' approach. It may be convenient for topological considerations, but perhaps less so in classical analysis. A function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:(X,O_X) \rightarrow (Y,O_Y)} is said to be continuous if for any open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \in O_Y} (respectively, closed subset of Y ) the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(U)=\{ x \in X \mid f(x) \in U\}} is an open set in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O_x} (respectively, a closed subset of X).

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