|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''Baire functions''' are [[function (mathematics)|function]]s obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by {{harvs|txt|authorlink=René-Louis Baire|first=René-Louis|last=Baire|year=1905}}. A [[Baire set]] is a set whose [[indicator function|characteristic function]] is a Baire function (not necessarily of any particular class, as defined below).
| | They call me Emilia. South Dakota is where me and my husband live. My day job is a meter reader. Body building is what my family and I appreciate.<br><br>Check out my page [http://www.videoworld.com/blog/211112 http://www.videoworld.com/blog/211112] |
| | |
| ==Classification of Baire functions==
| |
| Baire functions of class ''n'', for any countable [[ordinal number]] ''n'', form a vector space of real-valued functions defined on a [[topological space]], as follows.
| |
| | |
| *The Baire class 0 functions are the [[continuous function]]s.
| |
| | |
| *The Baire class 1 functions are those functions which are the [[pointwise convergence|pointwise limit]] of a [[sequence]] of Baire class 0 functions.
| |
| | |
| *In general, the Baire class ''n'' functions are all functions which are the pointwise limit of a sequence of functions of Baire class less than ''n''.
| |
| | |
| Some authors define the classes slightly differently, by removing all functions of class less than ''n'' from the functions of class ''n''. This means that each Baire function has a well defined class, but the functions of given class no longer form a vector space.
| |
| | |
| [[Henri Lebesgue]] proved that (for functions on the unit interval) each Baire class of a countable ordinal number contains functions not in any smaller class, and that there exist functions which are not in any Baire class.
| |
| | |
| ==Baire class 1==
| |
| Examples:
| |
| *The [[derivative]] of any differentiable function is of class 1. An example of a differentiable function whose derivative is not continuous (at ''x''=0) is the function equal to <math>x^2\sin(1/x)</math> when ''x''≠0, and 0 when ''x''=0. An infinite sum of similar functions (scaled and displaced by rational numbers) can even give a differentiable function whose derivative is discontinuous on a dense set. However, it necessarily has points of continuity, which follows easily from The Baire Characterisation Theorem (below; take ''K''=''X''='''R''').
| |
| *The function equal to 1 if ''x'' is an integer and 0 otherwise. (An infinite number of large discontinuities.)
| |
| *The function that is 0 for irrational ''x'' and 1/''q'' for a rational number ''p''/''q'' (in reduced form). (A dense set of discontinuities, namely the set of rational numbers.)
| |
| | |
| The Baire Characterisation Theorem states that a real valued function f defined on a [[Banach space]] X is a Baire-1 function if and only if for every non-empty closed subset K of X, the restriction of f to K has a point of continuity relative to the topology of K.
| |
| | |
| By another theorem of Baire, for every Baire-1 function the points of continuity are a [[comeager]] [[Gδ set|''G''<sub>δ</sub>]]set {{harv|Kechris|1995|loc=Theorem (24.14)}}.
| |
| | |
| ==Baire class 2==
| |
| Examples:
| |
| *An example of a Baire class two function on the interval [0,1] that is not of class 1 is the characteristic function of the rational numbers, <math>\chi_\mathbb{Q}</math>, also known as the [[Dirichlet function]]. It is discontinuous everywhere.
| |
| *Similarly, the function which is 1 if ''x'' is in the [[Cantor set]] and 0 otherwise. This function is 0 for an uncountable set of ''x'' values, and 1 for an uncountable set. It is discontinuous wherever it equals 1 and continuous wherever it equals 0.
| |
| | |
| ==Baire class 3==
| |
| Examples:
| |
| {{Expand section|date=November 2013}}
| |
| | |
| ==See also==
| |
| | |
| *[[Baire set]]
| |
| *[[Nowhere continuous function]]
| |
| | |
| ==References==
| |
| | |
| *{{citation|first=R. |last=Baire|title=Leçons sur les fonctions discontinues, professées au collège de France|publisher= Gauthier-Villars |year=1905|url=http://archive.org/detail/leconsdiscontinues00bairrich}}
| |
| *{{citation|first=Alexander S.|last=Kechris|author-link=Alexander S. Kechris|title=Classical Descriptive Set Theory|publisher=Springer-Verlag|year=1995}}
| |
| | |
| ==External links==
| |
| * [http://eom.springer.de/b/b015030.htm Springer Encyclopaedia of Mathematics article on Baire classes]
| |
| | |
| [[Category:General topology]]
| |
| [[Category:Real analysis]]
| |
| [[Category:Types of functions]]
| |
They call me Emilia. South Dakota is where me and my husband live. My day job is a meter reader. Body building is what my family and I appreciate.
Check out my page http://www.videoworld.com/blog/211112