Continuous function: Difference between revisions
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imported>Hendra I. Nurdin (Stub for continuous function) |
imported>Hendra I. Nurdin m (punctuation for clarity) |
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In [[mathematics]], 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 a '''continuous function''' if for every point <math>y \in Y</math> such that <math>y=f(x)</math> for some <math>x \in X</math> ''and'' for every [[open set]] <math>U_y \in O_y</math> containing ''y'', 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 [[mathematics]], 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 a '''continuous function''' if for every point <math>y \in Y</math> such that <math>y=f(x)</math> for some <math>x \in X</math>, ''and'' for every [[open set]] <math>U_y \in O_y</math> containing ''y'', 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>. | ||
Revision as of 03:26, 15 September 2007
In mathematics, a function f from a topological space to another topological space , usually written as , is a continuous function if for every point such that for some , and for every open set containing y, there exists an open set containing x such that . Here .
