An Infinitely Differentiable Function That Is Not Analytic

In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, with this article constructing a counterexample.

The function

Consider the function
f(x)=\left\{\begin{matrix}\exp(-1/x^2) & \mbox{if}\ x\neq 0 \\ \\ 0 & \mbox{if}\ x=0 \end{matrix}\right.
where x is a real number.

How it is ill-behaved

It is clear from the definition that f has derivatives of all orders at points a \in \mathbb{R} other than a = 0, since there it is the composition of smooth functions. At a = 0, all the derivatives are 0, which may be calculated by a careful induction proof that the difference quotient has the form
f^{(n + 1)}(0) = \lim_{x \to 0} \frac{f^{(n)}(x) - f^{(n)}(0)}{x} = \lim_{x \to 0} \frac{P(x)}{x^m} \exp\left(-\frac{1}{x^2}\right)
for some m \in \mathbb{N}, where P(x) is a polynomial function; such a limit is necessarily zero. Therefore, the Taylor series of f is
\sum_{n=0}^\infty {0\over n!}x^n = 0\neq f(x)
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, the analytic continuation of f to the complex plane has an essential singularity at the origin, and hence is not even continuous, much less analytic. Indeed, all holomorphic functions are analytic, so that the failure of f to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.

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

One of the most important applicatons of this function is the construction of so-called mollifiers, which are important in theories of generalized functions, like e.g. Laurent Schwartz's theory of distributions. 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.

External links

 

<< PreviousWord BrowserNext >>
vocal group hall of fame
irish fiction
mricourt
sabotage (album)
alan thornett
north caledonian football league
number theoretic transform
boystown, chicago
scottish junior football association
biograph theater
richard verrall
the pied pipers
uss archerfish (ssn 678)
beleg
timeline of hacker history
colorblind james experience
david gascoyne
kings of arnor
chieftains of the dnedain
kappa (letter)
rho (letter)
list of romanian language poets
psi (letter)
david strickland
british surrealist group
oak cliff, texas
caryopsis
sheffield winter gardens
fair deal
billy vaughn
papandreou
the mists of avalon
andni
codex runicus
virtualdub
lords of andni
virtualdubmod
james p. cannon
inverted swan
jzsef mindszenty
jerald and sandra tanner
dobsonian
linksruck
elmira, ontario