In mathematics, the Dirichlet function is the indicator function 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} of the set of rational numbers Q {\displaystyle \mathbb {Q} } , i.e. 1 Q ( x ) = 1 {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)=1} if x is a rational number and 1 Q ( x ) = 0 {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)=0} if x is not a rational number (i.e. is an irrational number). 1 Q ( x ) = { 1 x ∈ Q 0 x ∉ Q {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}
It is named after the mathematician Peter Gustav Lejeune Dirichlet. It is an example of a pathological function which provides counterexamples to many situations.
Topological properties
- The Dirichlet function is nowhere continuous.
Proof
- 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 numbers are dense 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.
- The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows: ∀ x ∈ R , 1 Q ( x ) = lim k → ∞ ( lim j → ∞ ( cos ( k ! π x ) ) 2 j ) {\displaystyle \forall x\in \mathbb {R} ,\quad \mathbf {1} _{\mathbb {Q} }(x)=\lim _{k\to \infty }\left(\lim _{j\to \infty }\left(\cos(k!\pi x)\right)^{2j}\right)} 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.4
Periodicity
For any real number x and any positive rational number T, 1 Q ( x + T ) = 1 Q ( x ) {\displaystyle \mathbf {1} _{\mathbb {Q} }(x+T)=\mathbf {1} _{\mathbb {Q} }(x)} . 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 R {\displaystyle \mathbb {R} } .
Integration properties
- The Dirichlet function is not Riemann-integrable on any segment of R {\displaystyle \mathbb {R} } despite being bounded because the set of its discontinuity points is not negligible (for the Lebesgue measure).
- The Dirichlet function has both an upper Darboux integral (namely, b − a {\displaystyle b-a} ) and a lower Darboux integral (0) over any bounded interval [ a , b ] {\displaystyle [a,b]} — but they are not equal if a < b {\displaystyle a<b} , so the Dirichlet function is not Darboux-integrable (and therefore not Riemann-integrable) over any nondegenerate interval.
- The Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral.
Proof
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.
- The Dirichlet function is Lebesgue-integrable on R {\displaystyle \mathbb {R} } and its integral over R {\displaystyle \mathbb {R} } is zero because it is zero except on the set of rational numbers which is negligible (for the Lebesgue measure).
See also
- Thomae's function, a variation that is discontinuous only at the rational numbers
References
"Dirichlet-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] https://www.encyclopediaofmath.org/index.php?title=Dirichlet-function ↩
Dirichlet Function — from MathWorld http://mathworld.wolfram.com/DirichletFunction.html ↩
Lejeune Dirichlet, Peter Gustav (1829). "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données". Journal für die reine und angewandte Mathematik. 4: 157–169. https://eudml.org/doc/183134 ↩
Dunham, William (2005). The Calculus Gallery. Princeton University Press. p. 197. ISBN 0-691-09565-5. 0-691-09565-5 ↩