Describe and explain a fundamental difference between the st

Describe and explain a fundamental difference between the structure of an infinitely differential function on R^2 and an analytic function on C. (You may use any

Solution

A function f(x), defined in a closed interval I= [a, b] is said to be analytic in this interval, if to every point x₀ belonging to I there corresponds a Taylor series with a positive radius of convergence, converging to f(x) in a neighborhood of x₀. Since the derivative of a Taylor series is a Taylor series with the same radius of convergence as the given series, and therefore converging uniformly in every closed interval containing x₀, and contained in the interval of convergence, an analytic function is infinitely differentiable in I.

An infinitely differentiable function in [a, b] will be said to be an ‘i. d. function” in [a, b]. It is well known, that an i. d. function in [a, b] is not necessarily analytic in this interval.

For instance the function f(x) defined by the equalities

f(x) =e^{-1/x^2}, if x not = 0,

f(0) = 0, is i. d. in every closed interval, but is not analytic in any interval containing the point x =0