In mathematics, the Blumberg theorem states that for any real function
It is named after its discoverer, the Russian-American mathematician Henry Blumberg.
For instance, the restriction of the Dirichlet function (the indicator function of the rational numbers
admits a continuous restriction on a dense subset of
(equipped with its usual topology) is a Blumberg space.
A counterexample was given in 1974 by Ronnie Levy, conditional on Luzin's hypothesis, that
[2] The problem was resolved in 1975 by William A. R. Weiss, who gave an unconditional counterexample.
It was constructed by taking the disjoint union of two compact Hausdorff spaces, one of which could be proven to be non-Blumberg if the Continuum Hypothesis was true, the other if it was false.
[3] 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,[note 1] and so only those restrictions that are continuous are (potentially) interesting.
Similarly uninteresting, the restriction of any function (continuous or not) to a single point or to any finite subset of
One case that is considerably more interesting is that of a non-continuous function
-valued functions defined on dense subsets is that a continuous extension to all of
if one exists, will be unique (there exist continuous functions defined on dense subsets of
Thomae's function, for example, is not continuous (in fact, it is discontinuous at every rational number) although its restriction to the dense subset
This raises the question: can such a dense subset always be found?
The Blumberg theorem answer this question in the affirmative.
− no matter how 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
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 or poorly behaved.
Imagine, for instance, defining a function
completely at random (so its graph would be appear as infinitely many points scattered randomly about the plane
); no matter how you ended up imagining it, the Blumberg theorem guarantees that even this function has some dense subset on which its restriction is continuous.