05 Mar '05 08:02

Describe a function from the reals to the reals such that the image of any interval intersects with every other interval.

Can such a function have an inverse?

Can such a function have an inverse?

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my head- Joined
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my head07 Mar '05 21:05

theoreticaly all quantum wave-forms are of this nature (to my understanding the range of the image is inversly proportional to the size of the interval examined.)*Originally posted by Acolyte***It means: if I pick any four numbers a<b, c<d, there'll be some number x between a and b such that f(x) lies between c and d.**

how about:

limit(a-->0) (x/a)-[a(x+1)]

infintie slopes and greatest integer functions are both mathmaticaly clumsy, but i belive this fits the specifications (in theory).- Joined
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Loughborough09 Mar '05 20:42

"(to my understanding the range of the image is inversly proportional to the size of the interval examined.)"*Originally posted by fearlessleader***theoreticaly all quantum wave-forms are of this nature (to my understanding the range of the image is inversly proportional to the size of the interval examined.)**

how about:

limit(a-->0) (x/a)-[a(x+1)]

infintie slopes and greatest integer functions are both mathmaticaly clumsy, but i belive this fits the specifications (in theory).

Huh? If A is a subset of B, then f(A) is a subset of f(B), so I don't see what you're trying to say.

"limit(a-->0) (x/a)-[a(x+1)]"

What do the square brackets mean? If they're a rounding function, this limit is not defined for x other than 0.- Joined
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09 Mar '05 23:05

Can't realy think of a function that would satisfy this describtion - but I see no reason why it should not be possible to construct one.*Originally posted by Acolyte***Describe a function from the reals to the reals such that the image of any interval intersects with every other interval.**

Can such a function have an inverse?

It will be difficult to ensure that it has an inverse.

If its not injective the inverse will not be well defined.

So eventhough we require that the image of any interval intersects with every other interval, we can not go for a function that is surjective in to R from any subinterval. We will have to look for a function where the image of any interval is dense in R.

If we could think of a function that reorders the real numbers in a way that seems completely random then that might be a candidate, but reordering that that is not wellordered seems difficult.- Joined
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my head11 Mar '05 12:26

in quantum mechanics the higher the resolution with which you examin a thing, the greater it varies from the overall appearence. thus, if examined on a mathmaticaly perfect level, all quamtum mechanical equations would be infinitly erratic. i may be misunderstanding it.**Huh? If A is a subset of B, then f(A) is a subset of f(B), so I don't see what you're trying to say.**

What do the square brackets mean? If they're a rounding function, this limit is not defined for x other than 0.

correct me if i'm wrong: what we are looking for is a function which includes all reals on every interval, no matter how small.

the [ ] are a greatest intiger funtion. prehaps the stated equation would be undifined, but i think it would fit the bill. any funtion that fits these standereds would have to have an infinite number of points for which it was undifined in every interval, at least if it was to have any sembilence of continuity.- Joined
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tinyurl.com/y9ls7wbl11 Mar '05 14:32

I've studied multivariable calculus and a little linear algebra and differential equations and I am clueless. This must be at least upper division undergraduate math stuff.*Originally posted by benkoboy***wow is this a type of math or something b/c im completly out of it and can anyone teach me what this is?**- Joined
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my head11 Mar '05 20:25

i dont think it's that complex, just weird.*Originally posted by AThousandYoung***I've studied multivariable calculus and a little linear algebra and differential equations and I am clueless. This must be at least upper division undergraduate math stuff.**

iamatiger has the best idea so far, it's similar to mine, except it dosn't require greatest integer functions, which are discontinueus.- Joined
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Loughborough11 Mar '05 22:34

Sounds like a good idea, but some explanation would help. For example, how do you know that |R| = |R/Q| ? Saying they're both uncountable isn't the same as saying they're the same size.*Originally posted by THUDandBLUNDER***f:R->R such that f(x)=g(x+Q)**

where

g:R->R = the bijection between R and the uncountably infinite set of coset representatives of the Abelian group R/Q

(which must exist by the Axiom of Choice).

All bijective functions have inverses.

I must admit I was thinking also of what is possible in a choice-free world. It's possible to give an explicit (read: choice-free) example of a function which takes every interval to a dense set. But can you*construct*such a function that is also bijective? If not, why not?- Joined
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12 Mar '05 04:181 edit

Firstly, I should say that the function I suggested was not my idea.*Originally posted by Acolyte***Sounds like a good idea, but some explanation would help. For example, how do you know that |R| = |R/Q| ? Saying they're both uncountable isn't the same as saying they're the same size.**

I must admit I was thinking also of what is po ...[text shortened]... ruct[/i] such a function that is also bijective? If not, why not?

*Originally posted by Acolyte***Sounds like a good idea, but some explanation would help. For example, how do you know that |R| = |R/Q| ? Saying they're both uncountable isn't the same as saying they're the same size.**

{R} is uncountable.

{Q} is countable.

Hence {R/Q} has the same cardinality as {R}.

The original question said 'describe', not 'construct'.

.