Continuous function: Difference between revisions

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In [[mathematics]], a '''continuous function''' is, intuitively speaking, a [[function]] whose "value" does not jump abruptly for any vanishingly "small" variation to its argument. Another way to think about a continuous 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.
#REDIRECT [[Continuity]]
 
==Formal definition==
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>. 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|neighbourhood]] of ''x'' and a neighbourhood of <math>y=f(x)</math>.
 
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 (respectively, closed) set <math>U \in O_Y</math> the set <math>f^{-1}(U)=\{ x \in X \mid f(x) \in U\}</math> is an open (respectively, closed) set in <math>O_x</math>. In this definition, a continuous function is simply a function which maps open sets to open sets or, equivalently, closed sets to closed sets.
 
The first definition corresponds to a generalization of the <math>\delta-\epsilon</math> argument which are usually taught in first year calculus courses to, among other things, define continuity for functions which map the real numbers to itself.
 
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