Quick/easy question about analytic (holomorphic) functions

[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)?

 

[Tac-Tics]

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

The term analytic function is often used interchangeably with “holomorphic function”. The fact that the class of complex analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis.

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).

 

[Office_Shredder]

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).

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.

 

[Petek]

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)?

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