I am asked to use the Weierstrass M-test to show that the following function is continuous on $A = \mathbb{R}\setminus \mathbb{Z}$
$$f(x) = \sum_{n=1}^\infty \frac{1}{x+n} + \frac{1}{x-n}$$
My problem to solve this exercise is indeed to find the suitable bound $M_n$ , since the function
$f_n(x) = \frac{1}{x+n} + \frac{1}{x-n}$ has two vertical asymptotes at $\pm n$