Difference between revisions of "Dirichlet-function"
From Encyclopedia of Mathematics
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| − | The function which is equal to one at the rational points and to zero at the irrational points. It is also defined by the formula: | + | The function $D(x)$ which is equal to one at the rational points and to zero at the irrational points. It is also defined by the formula: |
| − | $$ | + | $$D(x)=\lim_{m\to\infty}\lim_{n\to\infty}(\cos m!\pi x)^{2n},$$ |
and belongs to the second Baire class (cf. [[Baire classes|Baire classes]]). It is not Riemann integrable on any segment but, since it is equal to zero almost-everywhere, it is Lebesgue integrable. | and belongs to the second Baire class (cf. [[Baire classes|Baire classes]]). It is not Riemann integrable on any segment but, since it is equal to zero almost-everywhere, it is Lebesgue integrable. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | ====Comment==== | ||
| + | This function is [[Periodic function|periodic]], with any non-zero rational number as [[Period of a function|period]]. | ||
Latest revision as of 21:31, 18 November 2017
The function $D(x)$ which is equal to one at the rational points and to zero at the irrational points. It is also defined by the formula:
$$D(x)=\lim_{m\to\infty}\lim_{n\to\infty}(\cos m!\pi x)^{2n},$$
and belongs to the second Baire class (cf. Baire classes). It is not Riemann integrable on any segment but, since it is equal to zero almost-everywhere, it is Lebesgue integrable.
References
| [1] | I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) |
Comment
This function is periodic, with any non-zero rational number as period.
How to Cite This Entry:
Dirichlet-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet-function&oldid=42322
Dirichlet-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet-function&oldid=42322
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article