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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




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(Dixon 1991), illustrated above. This function is continuous at irrational

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and discontinuous at rational
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(although a small interval around an irrational point
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contains infinitely many rational points, these rationals will have very large denominators). When viewed from a corner along the line
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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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References

Ballone, F.A. "On Volterra Spaces." Masters thesis, Youngstown State University, Jun.2010.Beanland, K.; Roberts, J.W.; & Stevenson, C. "Modifications of Thomae"s Function và Differentiability." Amer. Math. Monthly 116, 531-535, 2009.Bruckner, A; Bruckner, J.; và Thomson, B. Elementary Real Analysis, 2nd ed.. Upper Saddle River, NJ: Prentice Hall, 2008.Dixon, R. Mathographics. New York: Dover, pp.177 & 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. Http://www.mathematicaguidebooks.org/.

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Dirichlet Function

Cite this as:

Weisstein, Eric W. "Dirichlet Function."From slovenija-expo2000.com--A slovenija-expo2000.com web Resource. Https://slovenija-expo2000.com/DirichletFunction.html