Continuous (Smooth) vs Differentiable versus Analytic

I had a lot of trouble distinguishing between continuous and differentiable and analytic functions…

Analytic

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 n-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

    \[f(x)=\exp\left(\frac{-1}{x^2}\right) \text{ if } x \neq 0\]

    \[f(0)=0\]

This function is continuous and infinitely many times differentiable in x=0. The Taylor series around this point is the constant function T(x)=0, so the Taylor series doesn’t converge to the function f(x) in the neighborhood of 0.

Complex Sequences and Analytic Approximations thereof

Suppose we know the values of a complex analytic function f at all x+iy, for x,y\in\mathbb{Z}. Can we uniquely determine f?

 

The Weierstrass factorization theorem asserts that for any sequence \{a_n\} of nonzero complex numbers with |a_n| \to \infty, there is a nontrivial analytic function f whose zero set is precisely \{a_n\}. Clearly we can enumerate all the nonzero points a_n of the integer lattice in such a way that |a_n| \to \infty, so there is a nontrivial analytic f vanishing at all those points. Then the function z f(z) is also not the zero function, but vanishes at all those points and 0 as well.<

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