Definition 1: The point $z_0$ is called a singular point of a function $f$ if $f$ fails to be analytic at $z_0$ but is analytic at some point in every neighborhood of $z_0$
Definition 2: A singular point $z_0$ is said to be isolated if, in addition,there is a deleted neighborhood of $z_0$ throughout which $f$ is analytic
Example: The function $$\frac{1}{\sin(\frac{\pi}{z})}$$
has the singular points $z=0$ and $z=\frac{1}{n} (n=\pm 1,\pm2,\cdots)$, all lying on the segment of the real axis from $z=-1$ to $z=1$. $\underline{Each\; singular\; point \;except\; z=0 \;is\; isolated.}$
I'm confusing this underlined term, because i know clearly $z=0$ is not an isolated singular point, since every neighborhood of $0$ contains another singular points. This is ok.
But how the numbers of the form $\frac{1}{n}$ is isolated ? If i take, $n=1000000000000000000000$, then how we guarantee the neighborhood of this $\frac{1}{n}$ does not contains some remaining singular points ?
I'm sorry, this is very elementary for others, but i'm confusing
Any help is appreciated !
