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2 How it is ill-behaved 3 How this is a good thing... |
One can show that f has derivatives of all orders at every point on the real number line including 0. To show this when x = 0, use L'Hopital's rule, mathematical induction, and some simple substitutions.
[Detail could be put here.]
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.
By multiplying this by any infinitely differentiable function one can get another infinitely differentiable function with prescribed behavior on the interval [a, b] of the real number line whose support is bounded. Only by showing the existence of functions with this sort of behavior can one be sure that Laurent Schwartz's theory of distributions (or "generalized functions") does not become vacuous for lack of test functions.
How this is a good thing...
...in negative terms
This example teaches us that functions of a real variable are sometimes ill-behaved in way to which functions of a complex variable are immune....in positive terms
Via a sequence of piecewise function definitions [Details could be put here.] one may construct from this function another function g(x) such that
and further, such that g has derivatives of all orders at every point.