In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Blumberg theorem states that for any
real function
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
there is a
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of
such that the
restriction of
to
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
. It is named after its discoverer, the Russian-American mathematician
Henry Blumberg.
Examples
For instance, the restriction of the
Dirichlet function
In mathematics, the Dirichlet function is the indicator function \mathbf_\Q of the set of rational numbers \Q, i.e. \mathbf_\Q(x) = 1 if is a rational number and \mathbf_\Q(x) = 0 if is not a rational number (i.e. is an irrational number).
\mathb ...
(the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s
) to
is continuous, although the Dirichlet function is
nowhere continuous in
Blumberg spaces
More generally, a Blumberg space is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
for which any function
admits a continuous restriction on a dense subset of
The Blumberg theorem therefore asserts that
(equipped with its usual topology) is a Blumberg space.
If
is a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
then
is a Blumberg space if and only if it is a
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
. 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
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
Or equivalently:
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
was true, the other if it was false.
[Weiss 1975, Weiss 1977.]
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,
[Every function that is not continuous can be restricted to some dense subset (specifically, its domain) on which its restriction is not continuous, so only those subsets on which its restriction continuous are interesting.] 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
In mathematics, the Dirichlet function is the indicator function \mathbf_\Q of the set of rational numbers \Q, i.e. \mathbf_\Q(x) = 1 if is a rational number and \mathbf_\Q(x) = 0 if is not a rational number (i.e. is an irrational number).
\mathb ...
) 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 (continuous or not) to a single point or to any finite subset of
(or more generally, to any
discrete subspace of
such as the integers
) will be continuous.
One case that is considerably more interesting is that of a non-continuous function
whose restriction to some
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
(of its domain) continuous.
An important fact about continuous
-valued functions defined on dense subsets is that a
continuous extension
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
to all of
if one exists, will be unique (there exist continuous functions defined on dense subsets of
such as
that cannot be continuously extended to all of
).
Thomae's function
Thomae's function is a real-valued function of a real variable that can be defined as:
f(x) =
\begin
\frac &\textx = \tfrac\quad (x \text p \in \mathbb Z \text q \in \mathbb N \text\\
0 &\textx \text
\end
It is named after Carl ...
, for example, is not continuous (in fact, it is discontinuous at rational number) although its restriction to the dense subset
of irrational numbers is continuous.
Similarly, every
additive function
In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer variable ''n'' such that whenever ''a'' and ''b'' are coprime, the function applied to the product ''ab'' is the sum of the values of the funct ...
that is not
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
(that is, not of the form
for some constant
) is a
nowhere continuous function
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f is a function from real numbers to real numbers, then f is nowhere continuous ...
whose restriction to
is continuous (such functions are the non-trivial solutions to
Cauchy's functional equation
Cauchy's functional equation is the functional equation:
f(x+y) = f(x) + f(y).\
A function f that solves this equation is called an additive function. Over the rational numbers, it can be shown using elementary algebra that there is a single fa ...
).
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 (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
Pathology is the study of disease. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in the context of modern medical treatme ...
or poorly behaved. Imagine, for instance, defining a function
by picking each value
completely at random (so its graph would 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 dense subset on which its restriction is continuous.
See also
*
*
*
*
*
Notes
Citations
References
*
*
*
"Variations on Blumberg's Theorem" Jack B. Brown, ''Real Analysis Exchange'' 9, #1 (1983/1984), pp. 123–137, , .
*
'Big' Continuous Restrictions of Arbitrary Functions, K. C. Ciesielski, M. E. Martínez-Gómez and J. B. Seoane-Sepúlveda, ''The American Mathematical Monthly'', 126, #6 (June–July 2019), pp. 547–552, .
* "Strongly non-Blumberg spaces", Ronnie Levy, ''General Topology and its Applications'', 4, #2 (June 1974), pp. 173–177, .
* "A solution to the Blumberg problem", William A. R. Weiss,
Bulletin of the American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society.
Scope
It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...
81, #5 (September 1975), pp. 957–958, .
* "The Blumberg problem", William A. R. Weiss, ''
Transactions of the American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...
'' 230 (June 1977), pp. 71–85, , .
*
*
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Theorems in real analysis
Theorems in topology