Consider the function f defined on R by f(x) =0 if x ≤ 0, f(x) = e−1/x2 if x > 0. Prove that f is indefinitely differentiable on R, and that f(n)(0) = 0 for all n ≥ 1. Conclude that f does not have a converging power series expansion Sumn=0to[infinity] anxn for x near the origin. [Note: This problem illustrates an enormous difference between the notions of real-differentiability and complex-differentiability.]
