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Series with fractional part of $nx$
I am working through Question 10 from Chapter 7 in Baby Rudin and I am very confused.
Letting (x) denote the fractional part of the real number $x$, consider the function $f(x)=\sum_{n=1}^{\infty} \frac{(nx)}{n^2}, \; x \in \mathbb{R}$ Find all discontinuities of $f$, and show that they form a countable dense set. Show that $f$ is nevertheless Riemann-integrable on every bounded interval.
I have looked at the solution (here) and my question is the following:
- Why is $f(x)$ discontinuous on all rationals and continuous on all irrationals?
If I can get a cogent explanation for this fact then I understand the question and the proof. Any help is greatly appreciated!