# Quick/easy question about analytic (holomorphic) functions

1. Oct 4, 2009

### AxiomOfChoice

Is saying "$f$ is differentiable" equivalent to saying "$f$ is analytic/holomorphic?"

Also, does it make sense to talk about functions being analytic/holomorphic at a POINT, or do we always need to talk about them being analytic in some NEIGHBORHOOD of a point (i.e., on an open set)?

2. Oct 5, 2009

### Tac-Tics

http://en.wikipedia.org/wiki/Holomorphic_function

Being holomorphic at a point is also the same as saying it's holomorphic in some neighborhood of that point. (Just like being differentiable or continuous at a point).

3. Oct 5, 2009

### Office_Shredder

Staff Emeritus
A function can actually be continuous or differentiable at a point without being continuous or differentiable in a neighborhood. One example is f(x)=x2 if x is rational, 0 if x is irrational.

4. Oct 5, 2009

### Petek

The function $f(z) = |z|^2$ has a derivative at z = 0, but not at any other point (verification is left as an exercise). Therefore, although the derivative exists at z = 0, the function isn't analytic at z = 0.

Petek