|Re: What would YOU call this?|
Message #5 Posted by Crawl on 23 Dec 2006, 6:47 a.m.,
in response to message #1 by Chuck
I'd call it a function, because "stay out of the complex" is an artifical human requirement, while analytic continuity is mathematically natural. I'd also say x / x = 1 at x = 0 (not that it's undefined), and 1 - 1 + 1 - 1 + 1 -... = 1/2, though.
If you want to have a function that has no domain, why not define a function that really has no domain? Or at least something weirder than the logarithm, which is a perfectly normal function, except that it's multi-valued. How about
f(x) = 1^x + 1^(2*x) + 1^(3*x) + 1^(4*x) + 1^(5*x) + ...
which diverges to infinity for all x. Or even use something non-mathematical.
x, if x loves 0
f(x) = x^2, if x doesn't love 0
f(x) could take values, but since it's hard to establish if one number loves another, it might be undefined for all x.