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Indicator function of rational numbers From Wikipedia, the free encyclopedia
In mathematics, the Dirichlet function[1][2] is the indicator function of the set of rational numbers , i.e. if x is a rational number and if x is not a rational number (i.e. is an irrational number).
It is named after the mathematician Peter Gustav Lejeune Dirichlet.[3] It is an example of a pathological function which provides counterexamples to many situations.
For any real number x and any positive rational number T, . The Dirichlet function is therefore an example of a real periodic function which is not constant but whose set of periods, the set of rational numbers, is a dense subset of .
Using an enumeration of the rational numbers between 0 and 1, we define the function fn (for all nonnegative integer n) as the indicator function of the set of the first n terms of this sequence of rational numbers. The increasing sequence of functions fn (which are nonnegative, Riemann-integrable with a vanishing integral) pointwise converges to the Dirichlet function which is not Riemann-integrable.
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