Is a function being analytic considered as a local property?

Question

Sorry for being pedantic... I was just wondering if analyticity of a complex function considered as a local property? Apparently differentiability is considered as a local property. But analyticity assumes differentiability not only at a point but in a neibourhood of that point as well. So would it be more than just a local property?

Answer 1

This is an interesting question. It really is not a "local property", since, if two analytic functions agree on an open set and have a common connected domain, they must agree globally.

On a non-connected domain things are a little different. Suppose you have two disjoint open disks in the plane and define a function $f$ to be $1$ on one disk and $2$ on the other. This function is analytic. However, this is a sort of artificial situation.

So, I would tend to think of the property as being a "global" one.

Answer 2

Analyticity is a local property -- as opposed to global and pointwise properties (I haven't encountered the latter type very often, but e.g. "being zero" is pointwise).

It is called that way because we don't need to know the behaviour of our function $f$ everywhere to be able to decide if it is analytic at a point. On the other hand, we cannot decide if $f$ is analytic at $z$ by looking at $f(z)$ alone; we need to know the power series of $f$, which can only be obtained by differentiation. For this operation, we need to know the behaviour of $f$ locally, in a neighbourhood around $f$. On the other hand, to decide analyticity at $z$, we needn't concern ourselves with what $f$ does at a reasonable distance to $z$, because this is independent of the derivatives of $f$ at $z$ -- just as in the real case.

I hope this clarifies matters a bit.