a pathological function of Riemann

The periodic mantissa functiontt-t  has at each integer value of t a jump (saltus) equal to -1, being in these points continuousMathworldPlanetmath from the right but not from the left.  For every real value t, one has

0t-t<1. (1)

Let us consider the series

n=1nx-nxn2 (2)

due to Riemann.  Since by (1), all values of  x  and  n+  satisfy

0nx-nxn2<1n2, (3)

the series is, by Weierstrass’ M-test, uniformly convergent on the whole (see also the p-test).  We denote by S(x) the sum functionMathworldPlanetmath of (2).

The nth term of the series (2) defines a periodic functionMathworldPlanetmath

xnx-nxn2 (4)

with the period (http://planetmath.org/PeriodicFunctions) 1n and having especially for  0x<1n  the value xn.  The only points of discontinuity of this function are

x=mn  (m=0,±1,±2,), (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  -1n2.

The theorem of the entry one-sided continuity by series   implies that the sum function S(x) is continuous in every irrational point x, because the series (2) is uniformly convergent for every x and its terms are continuous for irrational points x.

Since the terms (4) of (2) are continuous from the right in the rational points (5), the same theorem implies that S(x) is in these points continuous from the right.  It can be shown that S(x) is in these points discontinuousMathworldPlanetmath from the left having the jump equal to  -π26n2.


  • 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