Continuous function: Difference between revisions
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imported>Hendra I. Nurdin m (punctuation for clarity) |
imported>Hendra I. Nurdin (Corrected the definition) |
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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 | 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>x \in X</math> and 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>. | ||
Revision as of 07:31, 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 and for every open set containing the point y=f(x), there exists an open set containing x such that . Here .
