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You are not logged in. #1 20090121 04:17:19
Looking for a functionI found this question a while ago now, but haven't made much progress with it. (Or prove that such a function doesn't exist) Can anyone shed light on this? Edit: Just remembered, the question doesn't actually require you to find an example of such a function. Proving its existence would be enough. Why did the vector cross the road? It wanted to be normal. #2 20090121 06:54:41
Re: Looking for a functionDo we also need to assume the opposite, i.e. that If not the question is trivial: just let f(x) = c where c is any irrational number. Last edited by TheDude (20090121 06:57:12) Wrap it in bacon #3 20090121 07:02:31#4 20090121 07:04:39
Re: Looking for a functionDid you read my post carefully? Wrap it in bacon #5 20090121 07:06:26
Re: Looking for a functionLook.
Now stop asking stupid questions! Last edited by JaneFairfax (20090121 07:12:52) #6 20090121 07:10:29
Re: Looking for a functionYour statement comes out of my one by using contrapositions. so Replacing those rightarrows with left or double arrows is allowed, so by using double arrows we have that my condition is equivalent to yours. Unfortunately, that means your trivial solution won't work. Why did the vector cross the road? It wanted to be normal. #7 20090121 07:14:43
Re: Looking for a functionI figured it couldn't be that easy. Wrap it in bacon #8 20090121 07:37:40
Re: Looking for a functionPerhaps my first thought will be more helpful. The irrationals are uncountably infinite, and f(x) will have to map them only to the rationals, which are countably infinite. Someone more educated on the subject will have to either verify or refute this, but it seems to me that such a mapping must include at least one member of the set of rationals which has an uncountably infinite number of irrational numbers mapped to it. Let's call this number m. Last edited by TheDude (20090121 07:40:07) Wrap it in bacon #9 20090122 01:36:41
Re: Looking for a functionSuch a function may not exist, but if it does, then most likely it won't be expressible in closed form. Indeed I think such a function (if it exists) will not be differentiable (like the function discovered by Weierstrass). #10 20090122 02:26:36
Re: Looking for a functionThe Weierstraß function is a “pathological” case (as described by the Wikipedia article on it). I highly doubt that the function we’re looking for (if it exists) is going to be anything as “pathological” – otherwise people would have known about it much earlier and wouldn’t have had to wait until the late 19^{th} century for an everywherecontinuous nowheredifferentiable function to be discovered. #11 20090122 17:45:49
Re: Looking for a functionAnswer: http://planetmath.org/encyclopedia/ThereIsNoContinuousFunctionThatSwitchesTheRationalNumbersWithTheIrrationalNumbers.html "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #12 20090123 00:09:46
Re: Looking for a function<That link's too long for my browser to show it all, so, Link> Why did the vector cross the road? It wanted to be normal. #13 20090123 00:17:50
Re: Looking for a functionI love the proof. Instead of using analysis methods (as you would immediately think of doing at first) it uses topology methods. And I <3 topology more than analyss. 