Nowhere continuous
Encyclopedia
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 number
s to real numbers, then f(x) is nowhere continuous if for each point x there is an ε > 0 such that for each δ > 0 we can find a point y such that |x − y| < δ and |f(x) − f(y)| ≥ ε. Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.
More general definitions of this kind of function can be obtained, by replacing the absolute value
by the distance function in a metric space
, or by using the definition of continuity in a topological space
.
One example of such a function is the indicator function of the rational number
s, also known as the Dirichlet function, named after German
mathematician
Peter Gustav Lejeune Dirichlet. This function is written IQ and has domain
and codomain
both equal to the real number
s. IQ(x) equals 1 if x is a rational number
and 0 if x is not rational. If we look at this function in the vicinity of some number y, there are two cases:
In plainer terms, between any two irrationals, there is a rational, and vice versa.
The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
for integer j and k.
This shows that the Dirichlet function is a Baire class
2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a meagre set
.
In general, if E is any subset of a topological space
X such that both E and the complement of E are dense in X, then the real-valued function which takes the value 1 on E and 0 on the complement of E will be nowhere continuous. Functions of this type were originally investigated by Dirichlet
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a nowhere continuous function, also called an everywhere discontinuous function, is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
that is not continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
at any point of its domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
. If f is a function from real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s to real numbers, then f(x) is nowhere continuous if for each point x there is an ε > 0 such that for each δ > 0 we can find a point y such that |x − y| < δ and |f(x) − f(y)| ≥ ε. Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.
More general definitions of this kind of function can be obtained, by replacing the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
by the distance function in a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
, or by using the definition of continuity in a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
.
One example of such a function is the indicator function of the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, also known as the Dirichlet function, named after German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Peter Gustav Lejeune Dirichlet. This function is written IQ and has domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
and codomain
Codomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
both equal to the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. IQ(x) equals 1 if x is a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
and 0 if x is not rational. If we look at this function in the vicinity of some number y, there are two cases:
- If y is rational, then f(y) = 1. To show the function is not continuous at y, we need to find an ε such that no matter how small we choose δ, there will be points z within δ of y such that f(z) is not within ε of f(y) = 1. In fact, 1/2 is such an ε. Because the irrational numberIrrational numberIn mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
s are denseDense setIn topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in the reals, no matter what δ we choose we can always find an irrational z within δ of y, and f(z) = 0 is at least 1/2 away from 1. - If y is irrational, then f(y) = 0. Again, we can take ε = 1/2, and this time, because the rational numbers are dense in the reals, we can pick z to be a rational number as close to y as is required. Again, f(z) = 1 is more than 1/2 away from f(y) = 0.
In plainer terms, between any two irrationals, there is a rational, and vice versa.
The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
for integer j and k.
This shows that the Dirichlet function is a Baire class
Baire function
In mathematics, Baire functions are certain sets of functions. They are studied in several fields of mathematics, including real analysis andtopology....
2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a meagre set
Meagre set
In the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible...
.
In general, if E is any subset of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X such that both E and the complement of E are dense in X, then the real-valued function which takes the value 1 on E and 0 on the complement of E will be nowhere continuous. Functions of this type were originally investigated by Dirichlet
Johann Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...
.
See also
- Thomae%27s function (also known as the popcorn function) - a function that is continuous at all irrational numbers and discontinuous at all rational numbers.
External links
- Dirichlet Function -- from MathWorld
- The Modified Dirichlet Function by George Beck, The Wolfram Demonstrations Project.