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{{Short description|Any real function on R admits a continuous restriction on a dense subset of R}}
In [[mathematics]], the '''Blumberg theorem''' states that for any [[real function]] <math>f : \Reals \to \Reals</math> there is a [[Dense set|dense subset]] <math>D</math> of <math>\Reals</math> such that the [[Restriction_(mathematics)|restriction]] of <math>f</math> to <math>D</math> is [[continuous function|continuous]].

==Examples==

For instance, the restriction of the [[Dirichlet function]] (the [[indicator function]] of the [[rational number]]s <math>\Q</math>) to <math>\Q</math> is continuous, although the Dirichlet function is [[Nowhere continuous function|nowhere continuous]] in <math>\Reals.</math>

==Blumberg spaces==

More generally, a '''Blumberg space''' is a [[topological space]] <math>X</math> for which any function <math>f : X \to \Reals</math> admits a continuous restriction on a dense subset of <math>X.</math> The Blumberg theorem therefore asserts that <math>\mathbb{R}</math> (equipped with its usual topology) is a Blumberg space.

If <math>X</math> is a [[metric space]] then <math>X</math> is a Blumberg space if and only if it is a [[Baire space]].

==Motivation and discussion==

The restriction of any continuous function to any subset of its domain (dense or otherwise) is always continuous, so the conclusion of the Blumberg theorem is only interesting for functions that are not continuous. Given a function that is not continuous, it is typically not surprising to discover that its restriction to some subset is once again not continuous,<ref group=note>Every function <math>f</math> that is not continuous can be restricted to some dense subset <math>D</math> (specifically, its domain) on which its restriction <math>f\vert_D</math> is not continuous, so only those subsets on which its restriction {{em|is}} continuous are interesting.</ref> and so only those restrictions that are continuous are (potentially) interesting.
Such restrictions are not all interesting, however. For example, the restriction of any function (even one as interesting as the [[Dirichlet function]]) to any subset on which it is constant will be continuous, although this fact is as uninteresting as constant functions.
Similarly uninteresting, the restriction of {{em|any}} function (continuous or not) to a single point or to any finite subset of <math>\Reals</math> (or more generally, to any [[Discrete space|discrete subspace]] of <math>\Reals,</math> such as the integers <math>\Z</math>) will be continuous.

One case that is considerably more interesting is that of a non-continuous function <math>f</math> whose restriction to some [[Dense set|dense subset]] <math>D</math> (of its domain) {{em|is}} continuous.
An important fact about continuous <math>\Reals</math>-valued functions defined on dense subsets is that a [[continuous extension]] to all of <math>\Reals,</math> if one exists, will be unique (there exist continuous functions defined on dense subsets of <math>\Reals,</math> such as <math>f(x) = 1/x,</math> that cannot be continuously extended to all of <math>\Reals</math>).

[[Thomae's function]], for example, is not continuous (in fact, it is discontinuous at {{em|every}} rational number) although its restriction to the dense subset <math>\R\setminus\Q</math> of irrational numbers is continuous.
Similarly, every [[additive function]] <math>\Reals \to \Reals</math> that is not [[Linear map|linear]] (that is, not of the form <math>x \mapsto c x</math> for some constant <math>c \in \Reals</math>) is a [[nowhere continuous function]] whose restriction to <math>\Q</math> is continuous (such functions are the non-trivial solutions to [[Cauchy's functional equation]]).
This raises the question: can such a dense subset always be found? The Blumberg theorem answer this question in the affirmative.
In other words, every function <math>\R \to \R</math> − no matter how [[Pathological (mathematics)#Well-behaved|poorly behaved]] it may be − can be restricted to some dense subset on which it is continuous.
Said differently, the Blumberg theorem shows that there does not exist a function <math>\R \to \R</math> that is so poorly behaved (with respect to continuity) that all of its restrictions to all possible dense subsets are discontinuous.

The theorem's conclusion becomes more interesting as the function becomes more [[Pathological (mathematics)|pathological]] or poorly behaved. Imagine, for instance, defining a function <math>f : \Reals \to \Reals</math> by picking each value <math>f(x)</math> completely at random (so its graph would be appear as infinitely many points scattered randomly about the plane <math>\Reals^2</math>); no matter how you ended up imagining it, the Blumberg theorem guarantees that even this function has {{em|some}} dense subset on which its restriction is continuous.

==See also==

* {{annotated link|Closed graph theorem (functional analysis)}}
* {{annotated link|Densely defined operator}}
* {{annotated link|Hahn–Banach theorem}}
* {{annotated link|Tietze extension theorem}}
* {{annotated link|Whitney extension theorem}}

==Notes==

{{reflist|group=note}}

==References==

{{reflist}}

* {{cite journal|title=New properties of all real functions|first=Henry|last=Blumberg|journal=Proceedings of the National Academy of Sciences|volume=8|issue=1|year=1922|pages=283–288 |doi=10.1073/pnas.8.10.283 |pmid=16586898 |pmc=1085149 |url=http://www.pnas.org/content/8/10/283.full.pdf |doi-access=free }}
* {{cite journal|title=New properties of all real functions|first=Henry|last=Blumberg|journal=Transactions of the American Mathematical Society|volume=24|year=1922|issue=2 |page=113-128|doi=10.1090/S0002-9947-1922-1501216-9 |url=https://www.ams.org/journals/tran/1922-024-02/S0002-9947-1922-1501216-9}}
* {{cite journal|title=Metric spaces in which Blumberg's theorem holds|first1=J. C.|last1=Bradford|first2=Casper|last2=Goffman|journal=Proceedings of the American Mathematical Society|volume=11|year=1960|page=667-670|url=https://www.ams.org/journals/proc/1960-011-04/S0002-9939-1960-0146310-1}}
* {{cite journal|title=Topological spaces in which Blumberg's theorem holds|first=H. E.|last=White|journal=Proceedings of the American Mathematical Society|volume=44|year=1974|page=454-462|url=https://www.ams.org/journals/proc/1974-044-02/S0002-9939-1974-0341379-3}}
* https://www.encyclopediaofmath.org/index.php/Blumberg_theorem

{{Topology}}

[[Category:Theorems in real analysis]]
[[Category:Theorems in topology]]

Blumberg theorem - Revision history

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