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The Dirichlet function for distinct real numbers and (usually taken as and ) is defined by

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

The name "Dirichlet function" has also been applied to the related Thomae function, more specifically called the modified Dirichlet function (Bruckner et al. 2008). Unlike the function defined above, the Thomae function is continuous at every irrational number.

See also

Continuous Function, Dirichlet Beta Function, Dirichlet Eta Function, Dirichlet Lambda Function, Irrational Number, Rational Number, Thomae Function

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References

Bruckner, A.; Bruckner, J.; and Thomson, B. Elementary Real Analysis, 2nd ed. Upper Saddle River, NJ: Prentice Hall, 2008.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. https://www.mathematicaguidebooks.org/.

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Dirichlet Function

Cite this as:

Weisstein, Eric W. "Dirichlet Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DirichletFunction.html

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