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

(1)

Let us consider the series

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{nx-\lfloor nx\rfloor }{n^2}}$(2)

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

(3)

the series is, by Weierstrass’ M-test, uniformly convergent on the whole $ℝ$ (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

$x\mapsto {\displaystyle \frac{nx-\lfloor nx\rfloor }{n^2}}$(4)

with the period (http://planetmath.org/PeriodicFunctions) $\frac{1}{n}$ and having especially for    the value $\frac{x}{n}$.  The only points of discontinuity of this function are

$x={\displaystyle \frac{m}{n}}\mathit{\hspace{1em}\hspace{1em}}(m=0,\pm 1,\pm 2,\mathrm{\dots }),$(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  $-{\displaystyle \frac{{\pi }^2}{6n^2}}$.