Pri la Funkcia Ekvacio f(x + y) = f(x) + f(y)
Posted by Tom Leinster
Jam de longe, Cauchy pruvis ke kontinua funkcio kiu verigas la funkcian ekvacion kiuj ajn estu la nombroj , necese estas homogena, unuagrada funkcio .
So begins Maurice Fréchet’s 1913 paper in L’Enseignement Mathématique. I came across it when I was trying to find the right reference for the solution of this functional equation. Apparently Cauchy was the first to prove that when is continuous, the only solutions are for some constant . Mark Meckes had told me that Lebesgue measurability of was sufficient, and found a nice explanation at the Tricki.
But I needed to find the original reference. And when I tracked it down, I was surprised and intrigued to find that it was in Esperanto.
It turns out that Fréchet, apart from making numerous contributions to analysis, was a keen Esperantist, publishing many papers in the subject and (as Mark pointed out to me) serving as president of the Internacia Scienca Asocio Esperantista. I’m looking forward to citing it.
Incidentally, Fréchet’s result can be improved further. Mark Kormes, in 1926, showed that one only needs to assume to be bounded on some set of positive measure. (This is a weaker condition than measurability.) But, boringly, his paper is in English.
Re: Pri la Funkcia Ekvacio f(x + y) = f(x) + f(y)
Wow! Cool! But you should have posted this blog entry using rot13.
A cute fact, sort of obvious from what you said, is that starting with Zermelo-Fraenkel set theory, it’s perfectly consistent to assume either that does have solutions other than the obvious ones, or that it doesn’t. The Axiom of Choice implies it does have other solutions… which unfortunately we are unable to write down. The Axiom of Determinancy implies it doesn’t.
Guess which axiom most mathematicians prefer.
And another, slightly harder puzzle: guess which famous mathematician wrote sentences like this: