Dirichlet Function

Let c and d!=c be real numbers (usually taken as c=1 and d=0). The Dirichlet function is defined by

 D(x)={c   for x rational; d   for x irrational

and is discontinuous everywhere. The Dirichlet function can be written analytically as


Because the Dirichlet function cannot be plotted without producing a solid blend of lines, a modified version, sometimes itself known as the Dirichlet function (Bruckner et al. 2008), Thomae function (Beanland et al. 2009), or small Riemann function (Ballone 2010, p. 11), can be defined as

 D_M(x)={0   for x irrational; 1/b   for x=a/b a reduced fraction

(Dixon 1991), illustrated above. This function is continuous at irrational x and discontinuous at rational x (although a small interval around an irrational point x contains infinitely many rational points, these rationals will have very large denominators). When viewed from a corner along the line y=x in normal perspective, a quadrant of Euclid's orchard turns into the modified Dirichlet function (Gosper).

See also

Continuous Function, Dirichlet Beta Function, Dirichlet Eta Function, Dirichlet Lambda Function, Euclid's Orchard, Irrational Number, Rational Number

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Ballone, F. A. "On Volterra Spaces." Masters thesis, Youngstown State University, Jun. 2010.Beanland, K.; Roberts, J. W.; and Stevenson, C. "Modifications of Thomae's Function and Differentiability." Amer. Math. Monthly 116, 531-535, 2009.Bruckner, A; Bruckner, J.; and Thomson, B. Elementary Real Analysis, 2nd ed.. Upper Saddle River, NJ: Prentice Hall, 2008.Dixon, R. Mathographics. New York: Dover, pp. 177 and 184-186, 1991.Tall, D. "The Gradient of a Graph." Math. Teaching 111, 48-52, 1985.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, pp. 32-33, 2004.

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Dirichlet Function

Cite this as:

Weisstein, Eric W. "Dirichlet Function." From MathWorld--A Wolfram Web Resource.

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