Continuity: Difference between revisions

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imported>Aleksander Stos
imported>Aleksander Stos
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:<math>\lim_{x\to x_0} f(x)  = f(x_0).</math>
:<math>\lim_{x\to x_0} f(x)  = f(x_0).</math>


==Continuous function==
==Continuous function - global approach==
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'').

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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 definition of continuity

A function f from a topological space to another topological space , usually written as , is said to be continuous at the point if for every open set containing the point y=f(x), there exists an open set containing x such that . Here . In a variation of this definition, instead of being open sets, and can be taken to be, respectively, a neighbourhood of x and a neighbourhood of .

This definition is a generalization of the 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 if (it is defined in a neighborhood of and) for any there exist such that

Simply stated,

Continuous function - global approach

If the function f is continuous at every point 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 is said to be continuous if for any open set (respectively, closed subset of Y ) the set is an open set in (respectively, a closed subset of X).