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# a pathological function of Riemann

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

$\displaystyle 0\leqq t\!-\!\lfloor{t}\rfloor<1.$ | (1) |

$\displaystyle\sum_{{n=1}}^{\infty}\frac{nx\!-\!\lfloor{nx}\rfloor}{n^{2}}$ | (2) |

due to Riemann. Since by (1), all values of $x\in\mathbb{R}$ and $n\in\mathbb{Z}_{+}$ satisfy

$\displaystyle 0\;\leqq\;\frac{nx\!-\!\lfloor{nx}\rfloor}{n^{2}}\;<\;\frac{1}{n% ^{2}},$ | (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

$\displaystyle x\mapsto\frac{nx\!-\!\lfloor{nx}\rfloor}{n^{2}}$ | (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

$\displaystyle x\;=\;\frac{m}{n}\qquad(m=0,\,\pm 1,\,\pm 2,\,\ldots),$ | (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 $\displaystyle-\frac{\pi^{2}}{6n^{2}}$.

# References

- 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.2. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).

## Mathematics Subject Classification

40A05*no label found*26A15

*no label found*26A03

*no label found*

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