It is well known that the function defined by
$$g(x) = \sum_{n\geq 1 } \sin(nx )/n $$
is a piece-wise linear function $x$, and has jumps at $x = 2 m \pi $. Its picture is
Now let us consider a modified function
$$f(x) = \sum_{n\geq 1 } \sin((n+\alpha /n)x )/n ,$$
where $\alpha $ is some real number. For $\alpha = 0.1$, its picture is
It also has jumps at $x = 2 m \pi $.
I am trying to understand it. It is not a standard Fourier series. How can we prove that it has discontinuities at $x = 2 m \pi$? Apparently, we have
$$ f(2m \pi ) = \frac{1}{2} (f(2 m \pi +0^+ ) + f(2 m\pi +0^- )). $$
This is also an interesting point.
I think I understand the phenomenon now. It is simple. Just consider the difference of the two series
$$ f- g = \sum_{n\geq 1} \frac{2}{n} \cos((n + \alpha /2n ) x)\sin (\alpha x /2 n ) . $$
This series converges uniformly into a continuous function.