discontinuity of $f(x) = \sum_{n\geq 1} \sin((n+ \alpha /n)x)/n$ 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. 
