Nowhere continuous function
The Dirichlet function ( after the German mathematician Peter Gustav Lejeune Dirichlet, sometimes referred to as Dirichlet jump function ) is a mathematical function, which is commonly referred to. It is the characteristic function of the rational numbers, and thus defined as:
Properties
The Dirichlet function is an example of
- A discontinuous at any point in its domain of definition function
- A function of the second class in the classification of Baire:
- A Lebesgue - integrable function, but which is not Riemann integrable.
Riemann integrability
The Dirichlet function is Riemann integrable in any proper interval, since for every partition in the subinterval always be both rational and irrational numbers, and thus
Always 0 ( because the infimum is always 0) and
Always the length of the interval over which is integrated is (because the supremum is always 1 and is thus simply adds the length of each sub-intervals ).
But Riemann integrability requires just the equality, so that applies:
But do not converge for any decomposition lower and upper sums to the same value, is Riemann integrable on any interval.
Lebesgue integrability
Since the Dirichlet function is a simple function, ie a measurable function that takes only finitely many values that are not to even negative, the Lebesgue integral over any interval can be written as follows:
Where stands for the Lebesgue measure.
For any value of results from the multiplication by 0, the result 0 This is due to a convention in measure theory, even if the other factor is infinite. In contrast, is always 0 because the point set of rational numbers is countable, and thus is a measure zero.
The overall result is thus for the Dirichlet function in each interval:
Related function
A related function is defined as follows:
She is at every rational point of its domain discontinuous and continuous at every irrational point and in contrast to the Dirichlet function also Riemann integrable:
It is called, among other things about Thoma ash function.