I had a lot of trouble distinguishing between continuous and differentiable and analytic functions…
An analytic function is a function that is smooth (in the sense that it is continuous and infinitely times differentiable), and the Taylor series around a point converges to the original function in the neighbourhood of that point.
Continuous – but NOT analytic (taylor series does not converge)
A smooth function is a continuous function with a continuous derivative. Some texts use the term smooth for a continuous function that is infinitely many times differentiable (all the -th derivatives are thus continuous, since differentiability implies continuity). The existence of all derivatives doesn’t imply that the Taylor series converges. A famous example is the function
This function is continuous and infinitely many times differentiable in . The Taylor series around this point is the constant function , so the Taylor series doesn’t converge to the function in the neighborhood of .
Complex Sequences and Analytic Approximations thereof
Suppose we know the values of a complex analytic function at all , for . Can we uniquely determine ?
The Weierstrass factorization theorem asserts that for any sequence of nonzero complex numbers with , there is a nontrivial analytic function whose zero set is precisely . Clearly we can enumerate all the nonzero points of the integer lattice in such a way that , so there is a nontrivial analytic vanishing at all those points. Then the function is also not the zero function, but vanishes at all those points and 0 as well.<