The size of the neighbourhood of a point

Question

I was recently studying complex analysis, and read about analytic functions, some definitions of analytic functions were like functions differentiable at a point a and it’s neighbourhood, now what is essentially a neighbourhood, I mean is it the whole surrounding set or a particular sized circle, square surrounding the given point, what is its importance ?

Answer 1

One way to think about it is that everything coming from the outside world has to penetrate your neighborhood before arriving at your point. A neighborhood guarantees that there exists some tiny $\epsilon>0$ where any point $y$ where $|y-x|<\epsilon$ also lives inside your neighborhood. So any infinite sequence of points that gets closer and closer to your $x$ will eventually be entirely inside your neighborhood, where everyone is behaving nicely.

If I know, for example, that every point with 1/10000 of me is also a place where $f$ is differentiable, I am confident that there cannot be a sequence of non-differentiable points that "comes within 1/10000" of my home.

Answer 2

A neighborhood can be arbitrarily (as much as needed) small. How I think of it is not so much the size of your neighborhood that matters, but moreso the fact that such a neighborhood exists. This is because if you have some small neighborhood where the property holds, you can zoom in far enough to make things "big enough" without being interrupted, so to speak. If something does not hold in any neighborhood, you can zoom in as much as you want, and still have issues/counterexamples. This is all very non-rigorous and hand wavy, but it is how I like to think of things.