Analyicity versus continuity

Hello,

I am learning complex integration and differentiation at the moment, but I have yet to understand what an analytical function is and what a continuous function is. I feel it has something to do with continuous derivatives, whatever that means!

Are analyticity and continuity one and the same thing? Can a function be one and not the other?

Any help would be much appreciated

Regards,

Peter

quasar987
Homework Helper
Gold Member
Are you attempting to learn complex analysis before real analysis? Or even before calculus? This is not recommended, but if you must know, analicity and continuity are two different but related things. The relation between them is that an analytic function is necessarily continuous (but a function can be continuous without being analytical).

HallsofIvy
The simplest definition of analytic is "a function, f, is continuous at $z_0$ if and only if there exist some neighborhood such that the Taylor's series for f exists and converges to f(z) in that neighborhood."
There exist many continuous functions that are not analytic. A simple definition in functions of real numbers is |x| and for complex numbers |z|= $\sqrt{zz*}$ where z* is the complex conjugate of z.