I’ve been puzzling over this for weeks: find a funtion such that any interval on the funtion maps to all the reals. Just for clarification (it was my first mistake) the funtion should be independent of the interval.
Something to do with Pascal’s Triangle, perhaps?
Just a guess.
Does it have to do with the decimal expansion of the members of the interval?
This is also mostly a guess.
y = z*x Where you plot it on 2-D and z is anything? Yeah, it’s cheating.
Whoops, that’s function, not funtion. Ahh the over reliance on spell check.
Great suggestions, keep 'em coming!
You need to figure out how to divide the real line into an uncountable number of dense subsets. Then assign one such set to each real number. The function should take on the value x exactly on the set corresponding to x, for each x. There are a couple of different ways to construct all those dense subsets.
Gotta be careful doing this, or the Continuum Hypothesis is going to bite you in the butt. You have to make sure you have at least C such subsets; Aleph-1 might not be enough.
(If you have no idea what I’m talking about, that’s because it’s really obscure!)
Good point, MrDeath. However, I don’t think it’s really such a concern after all. For that to be a problem, you would have to be able to construct an uncountable set with cardinality strictly smaller than c, and we all know you can’t do that.
Define f1(x) = 1 / (2^2 *(x-1/2)) on [0,1], except f1(1/2) = 0
f2(x) = f1(x) - 1/(2^4 * (x-1/4)) - 1/(2^4 * (x-3/4))
f3(x) = f2(x) + 1/(2^6 * (x-1/8)) + 1/(2^6 * (x-3/8)) + …
etc.
(always substitute 0 for the corresponding term when x is exactly on a pole. e.g. the last term in f2(x) is dropped if x = 3/4).
Then define F(x) = lim (N -> infinity) fN(x)
Finally, the function would be G(x) = F(x - )
where is the largest integer less than x.