What is the difference between analyticity and continuity in functions?

[peterjaybee]

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]

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."
Analyticity implies continuity and, in fact, continuity of all derivatives.

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.

 

[slider142]

At its most basic, the motivation for continuity in real functions from R into R is "I can draw the entire graph of this function without lifting my pencil from the paper." Since we are not interested in the limitations/physics of pencils but in the behavior of the values in the range of the function when we vary variables in the domain of the function, we come up with a definition of continuity that basically says "when we move towards the point a₀ in the domain of a function f, we also move towards the point f(a₀) in the range of f." We then call f continuous at a₀. If f is continuous on its entire range, we say f is continuous.
You can easily generalize this to multivariable and vector functions, where we basically are interested in functions without holes or jumps. Be careful with pathological behavior, however! there are simple functions that are only continuous at a single point, and there are functions that are mathematically continuous that do not look like anything you could ever draw with a pencil (ie., Weirstrauss).
Generally, we notice that functions behave "nicer" when they have more continuous derivatives. Ie., a functions which has continuous derivatives of the 5th order (any polynomial) is said to be smoother than a function that has discontinuous derivatives of the 3rd order (ie., f(x) = {x², x >= 0; -x², x < 0} ).
Functions that have continuous derivatives of all orders are generally said to be smooth, like polynomials. There are non-polynomial functions that behave almost as nicely as polynomials: it is well known that many functions are equivalent to their Taylor series on some neighborhood around the point that the series is based on. Functions that are equivalent to their Taylor series (kind of like an infinite polynomial) everywhere (technically, on some neighborhood of each point) are said to be analytic.
These definitions, used in real analysis, motivate the definitions in complex analysis.