a pathological function of Riemann
Let us consider the series
due to Riemann. Since by (1), all values of and satisfy
The term of the series (2) defines a periodic function
with the period (http://planetmath.org/PeriodicFunctions) and having especially for the value . The only points of discontinuity of this function are
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 .
- 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.2. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
|Title||a pathological function of Riemann|
|Date of creation||2013-03-22 18:34:17|
|Last modified on||2013-03-22 18:34:17|
|Last modified by||pahio (2872)|
|Synonym||example of semicontinuous function|