robison@uiucdcsb.UUCP (01/24/85)
/* Written 11:01 am Jan 21, 1985 by chris@umcp-cs in uiucdcsb:net.math */
/* ---------- "Re: Re: Non-linear systems: discont" ---------- */
[I've moved this from net.physics]
Speaking of discontinous functions ...
One of my favorite functions is
{ 1/q, x rational and expressed as p/q in lowest terms
f(x) = {
{ 0, x irrational
This thing is continuous nowhere, yet differentiable everywhere.
(f'(x) = 0 for all x.)
Does anyone else have a favorite weird function that is also simple to
define?
--
(This line accidently left nonblank.)
In-Real-Life: Chris Torek, Univ of MD Comp Sci Dept (+1 301 454 7690)
UUCP: {seismo,allegra,brl-bmd}!umcp-cs!chris
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/* End of text from uiucdcsb:net.math */robison@uiucdcsb.UUCP (01/24/85)
My copy of the basenote was a mistake. Ignore it. - robison @ uiucdcs
bstewart@bnl.UUCP (Hugh Bruce Stewart) (02/13/85)
> > Speaking of discontinous functions ... > > One of my favorite functions is > > { 1/q, x rational and expressed as p/q in lowest terms > f(x) = { > { 0, x irrational > Anyone with an interest in such things might want to look at the book Counterexamples in Analysis by Gelbaum and Olmstead, Holden-Day, 1964. This is a unique compendium of all the things that can go wrong if theorems about continuity, integration, convergence, etc. are misused. Chapter 8 in particular shows why Lebesgues's definition of a definite integral is useful. One might say that his definition is made to handle functions such as density distributions on deterministically defined fractal sets. In a nutshell, Lebesgue's idea was to approximate the area under the curve f(x) by summing over the f(x)-axis instead of over the x-axis as Cauchy and Riemann had done. Bruce Stewart, Applied Math. Dept., Brookhaven National Lab.