I know that if a function $f$ fails to be analytic at $z_0$ but is analytic at some point in every neighborhood of $z_0$, then $z_0$ is a singular point.
But what is an isolated singular point?
Here is the definition from my book: A singular point $z_0$ is said to be isolated if, in addition [to being singular], there is a deleted neighborhood $0<|z-z_0|<\epsilon$ of $z_0$ throughout which $f$ is analytic.
I don't fully understand what a deleted neighborhood means.
Any clarification is appreciated.
Thanks.
Best Answer
Deleted Neighborhood
The proper name for a set such as ${x: 0 < |x – a| < δ}$. Deleted neighborhoods are encountered in the study of limits. It is the set of all numbers less than δ units away from a, omitting the number a itself.
Using interval notation the set ${x: 0 < |x – a| < δ} $ would be $(a – δ, a) ∪ (a, a + δ).$ In general, a deleted neighborhood of $a$ is any set $(c, a) ∪ (a, d)$ where $c < a < d.$
For example, one deleted neighborhood of $2$ is the set ${x: 0 < |x – 2| < 0.1}$, which is the same as $(1.9, 2) ∪ (2, 2.1).$