And here is the plot of indefinite sum of tan(x):

![alt text][1]


  [1]: http://static.itmages.ru/i/10/1020/h_1287588222_873cd3aff3.png

Here you can see tan(x) in red and its indefinite sum is in blue.

As you can see, the indefinite sum is fairly continuous. Oleg Eroshkin's conclusion that this function should be discontinuous everywhere apparently came from a false assumption that indefinite sum of a periodic function should also be periodic.

Though it is true that as $|x|$ grows the density of the poles grows, showing the same behavior as in function $f(x)=\tan(x^2)$