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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
a pathological function of Riemann
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(Example)
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The periodic mantissa function $t \mapsto t\!-\!\lfloor{t}\rfloor$ has at each integer value of $t$ a jump (saltus) equal to $-1$ , being in these points continuous from the right but not from
the left. For every real value $t$ , one has
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(1) |
Let us consider the series
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(2) |
due to Riemann. Since by (1), all values of $x \in \mathbb{R}$ and $n \in \mathbb{Z}_+$ satisfy
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(3) |
the series is, by Weierstrass' M-test, uniformly convergent on the whole $\mathbb{R}$ (see also the p-test). We denote by $S(x)$ the sum function of (2).
The $n^\mathrm{th}$ term of the series (2) defines a periodic function
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(4) |
with the period $\frac{1}{n}$ and having especially for $0 \leqq x < \frac{1}{n}$ the value $\frac{x}{n}$ . The only points of discontinuity of this function are
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(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 $\displaystyle-\frac{1}{n^2}$ .
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 discontinuous from the left having the jump equal to $\displaystyle-\frac{\pi^2}{6n^2}$ .
- 1
- E. LINDELÖF: Differentiali- ja integralilasku ja sen sovellutukset III.2. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
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"a pathological function of Riemann" is owned by pahio.
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Cross-references: discontinuous, rational, irrational, continuous, implies, one-sided continuity by series, theorem, vanishes, function, periodic function, sum function, p-test, uniformly convergent, Weierstrass M-test, Riemann, series, real, continuous from the right, points, saltus, jump, integer, mantissa function, periodic
This is version 6 of a pathological function of Riemann, born on 2008-12-01, modified 2008-12-28.
Object id is 11295, canonical name is APathologicalFunctionOfRiemann.
Accessed 498 times total.
Classification:
| AMS MSC: | 26A03 (Real functions :: Functions of one variable :: Foundations: limits and generalizations, elementary topology of the line) | | | 26A15 (Real functions :: Functions of one variable :: Continuity and related questions ) | | | 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences) |
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Pending Errata and Addenda
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