Table of contents |
2 How it is ill-behaved 3 How this is negatively a good thing 4 How this is positively a good thing |
Consider the real function
One can show that f has derivatives of all orders at every point including 0. To show this at x = 0, use L'Hopital's rule, mathematical induction, and some simple substitutions.
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But in proving this, one will find that f(n)(0) = 0 for all n. Therefore, the Taylor series of f is
The function
How it is ill-behaved
unless x = 0. Consequently f is not analytic at 0. This pathology cannot occur with functions of a complex variable rather than of a real variable.