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The function
Consider the real function
- <math>f(x)=\left\{\begin{matrix}\exp(-1/x^2) & \mbox{if}\ x\neq 0 \\ \\ 0 & \mbox{if}\ x=0 \end{matrix}\right\}.<math>
How it is ill-behaved
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, note that the limit from the left of (f(x + h) − f(x))/h at 0 is 0, and similarly for any f (n) we prove to exist. For any differentiable function R(x) the derivative for x > 0 of R(x)f(x) is
- <math>R'(x)\exp(-1/x^2) + R(x)(1/2x^3)\exp(-1/x^2) = \left[R'(x) + R(x)(1/2x^3)\right]\exp(-1/x^2)<math>.
If R is rational, then exp(−1/x2) decreases faster than 1/R, so the limit from the right at 0 is 0.
Thus f (n)(0) = 0 for all n. Therefore, the Taylor series of f is
- <math>\sum_{n=0}^\infty {0\over n!}x^n = 0\neq f(x)<math>
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. Note that although this function has derivatives of all orders over the real line, an analogous function defined over the complex plane would fail to even be continuous at z=0 and so is not holomorphic, in accordance with the theorem that holomorphic functions are analytic.
How this is a good thing...
...in negative terms
This example teaches us that functions of a real variable are sometimes ill-behaved in ways 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
- <math>g(x)=\left\{\begin{matrix}0 & \mbox{if} & x
1 & \mbox{if} & ab+\varepsilon \end{matrix}\right\}<math>
and further, such that g has derivatives of all orders at every point.
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.
The existence of these functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated by saying that the sheaf of differentiable functions on a differentiable manifold is flasque, in contrast with the analytic case.
The function above is generally used to build up partitions of unity on differentiable manifolds.
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