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Dirichlet Function

Let $c$ and $d\not=c$ be Real Numbers (usually taken as $c=1$ and $d=0$). The Dirichlet function is defined by

c & for $x$\ rational\cr
d & for $x$\ irrational.}

The function is Continuous at Irrational $x$ and discontinuous at Rational points. The function can be written analytically as

D(x)=\lim_{m, n\to\infty} \cos[(m!\pi x)^n].


Because the Dirichlet function cannot be plotted without producing a solid blend of lines, a modified version can be defined as

0 & for $x$\ irrational\cr
b & for $x=a/b$\ with $a/b$\ a reduced fraction\cr}

(Dixon 1991), illustrated above.

See also Continuous Function, Irrational Number, Rational Number


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.

© 1996-9 Eric W. Weisstein