This one is pretty involved so mad props to whoever can help me figure it out. I've been thinking about this for more than an hour and it's bugging me. Consider the function f(z) = 1/(sin(pi/z)). It has singular points at z=0 and z=1/n (where n is an integer). However, my book says each singular point except z=0 is isolated. (An isolated singular point is one where every epsilon neighborhood around that point is analytic.) Anyways, the argument is that if we find any epsilon neighborhood around z=0, then we can always find a 1/n (where n is an integer) inside this neighborhood. As we know, 1/n is a singular point hence there doesn't exist a neighborhood around z=0 that is analytic. This I understand. However, I don't get why this same argument couldn't be applied to the other singular points z=1/n. Namely, for any epsilon neighborhood around z=1/n, we can find some other z=1/m inside this neighborhood... can't we? My OCD prevents me from moving on forward in the chapter until I've figured this out exactly. It's driving me nuts. My prof doesn't have office hours until Tuesday of next week and he's notorious for not answering emails. If any of yall can help me figure this out before then, it'd be great.