a pathological function of Riemann
The periodic mantissa function has at each integer value of a jump (saltus) equal to , being in these points continuous from the right but not from the left. For every real value , one has
(1) |
Let us consider the series
(2) |
due to Riemann. Since by (1), all values of and satisfy
(3) |
the series is, by Weierstrass’ M-test, uniformly convergent on the whole (see also the p-test). We denote by the sum function of (2).
The term of the series (2) defines a periodic function
(4) |
with the period (http://planetmath.org/PeriodicFunctions) and having especially for the value . The only points of discontinuity of this function are
(5) |
where it vanishes and where it is continuous from the right but not from the left; in the point (5) this function apparently has the jump .
The theorem of the entry one-sided continuity by series implies that the sum function is continuous in every irrational point , because the series (2) is uniformly convergent for every and its terms are continuous for irrational points .
Since the terms (4) of (2) are continuous from the right in the rational points (5), the same theorem implies that is in these points continuous from the right. It can be shown that is in these points discontinuous from the left having the jump equal to .
References
- 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.2. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
Title | a pathological function of Riemann |
---|---|
Canonical name | APathologicalFunctionOfRiemann |
Date of creation | 2013-03-22 18:34:17 |
Last modified on | 2013-03-22 18:34:17 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 9 |
Author | pahio (2872) |
Entry type | Example |
Classification | msc 40A05 |
Classification | msc 26A15 |
Classification | msc 26A03 |
Synonym | example of semicontinuous function |
Related topic | DirichletsFunction |
Related topic | ValueOfTheRiemannZetaFunctionAtS2 |