Blumberg theorem

In mathematics, the Blumberg theorem states that for any real function there is a dense subset of such that the restriction of to is continuous. It is named after its discoverer, the Russian-American mathematician Henry Blumberg.

Examples

For instance, the restriction of the Dirichlet function to is continuous, although the Dirichlet function is nowhere continuous in

Blumberg spaces

More generally, a Blumberg space is a topological space for which any function admits a continuous restriction on a dense subset of The Blumberg theorem therefore asserts that is a Blumberg space.
If is a metric space then is a Blumberg space if and only if it is a Baire space. The Blumberg problem is to determine whether a compact Hausdorff space must be Blumberg. A counterexample was given in 1974 by Ronnie Levy, conditional on Luzin's hypothesis, that 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.

Motivation and discussion

The restriction of any continuous function to any subset of its domain 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, and so only those restrictions that are continuous are interesting.
Such restrictions are not all interesting, however. For example, the restriction of any 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 function to a single point or to any finite subset of will be continuous.
One case that is considerably more interesting is that of a non-continuous function whose restriction to some dense subset continuous.
An important fact about continuous -valued functions defined on dense subsets is that a continuous extension to all of if one exists, will be unique.
Thomae's function, for example, is not continuous although its restriction to the dense subset of irrational numbers is continuous.
Similarly, every additive function that is not linear is a nowhere continuous function whose restriction to is continuous.
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 − 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 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 by picking each value completely at random ; no matter how you ended up imagining it, the Blumberg theorem guarantees that even this function has dense subset on which its restriction is continuous.