[math-fun] [0, 1] to (0, 1)?
More of an intellectual curiosity question than anything useful. Is there a "nice" function that is one-to-one and continuous that maps the closed interval [0, 1] to the open interval (0, 1)? Ideally, such a "nice" function would be not a piecewise function or a power series and something that could be implemented with standard library functions. If such a thing doesn't exist, is there at least a "not nice" function that is one-to-one and maps the intervals? If not, why not? Thanks for any pointers or wisdom, Kerry Mitchell -- lkmitch@gmail.com www.fractalus.com/kerry
It is well-known that [i.e., I think it's true but can't be bothered to look up the reference] the pre-image of an open set under a continuous function is also open [note that it doesn't work in the opposite direction!]. See any elementary text on point-set topology. So your "nice" function does not exist. As to a succinct definition of a bijection, nothing suggests itself right now ... Anybody else? WFL On 11/30/06, Kerry Mitchell <lkmitch@gmail.com> wrote:
More of an intellectual curiosity question than anything useful. Is there a "nice" function that is one-to-one and continuous that maps the closed interval [0, 1] to the open interval (0, 1)? Ideally, such a "nice" function would be not a piecewise function or a power series and something that could be implemented with standard library functions. If such a thing doesn't exist, is there at least a "not nice" function that is one-to-one and maps the intervals? If not, why not?
Thanks for any pointers or wisdom, Kerry Mitchell -- lkmitch@gmail.com www.fractalus.com/kerry
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About the best you can do is to choose two sequences of numbers to map 0 and 1 to, and use the identity for everything else. For example, f(1/3^n) = 1/3^(n+1), f(1-1/3^n) = 1 - 1/3^(n+1), both for n>=0, all other f(x) = x. Franklin T. Adams-Watters -----Original Message----- From: lkmitch@gmail.com More of an intellectual curiosity question than anything useful. Is there a "nice" function that is one-to-one and continuous that maps the closed interval [0, 1] to the open interval (0, 1)? Ideally, such a "nice" function would be not a piecewise function or a power series and something that could be implemented with standard library functions. If such a thing doesn't exist, is there at least a "not nice" function that is one-to-one and maps the intervals? If not, why not? Thanks for any pointers or wisdom, Kerry Mitchell -- lkmitch@gmail.com www.fractalus.com/kerry ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.
On 30 Nov 2006 at 16:55, Kerry Mitchell wrote:
More of an intellectual curiosity question than anything useful. Is there a "nice" function that is one-to-one and continuous that maps the closed interval [0, 1] to the open interval (0, 1)? ... If such a thing doesn't exist, is there at least a "not nice" function that is one-to-one and maps the intervals? If not, why not?
As has been mentioned, there's no 'nice' way of doing it [in either direction, open->closed or vice versa]. But there *is* a pretty ugly way of doing it [mindbogglingly non constructive, but I suppose it is a 'function' in some sense of the term :o)]. Take any irrational # -- $PI will do fine: For n integer, f(n) = fp($PI * n) ;, n = 1, 2, 3, ... Since PI is irrational, this is a 1-1 "into" function. Let A be the range of f, and B = (0,1] - A. Then define, for r R if r = 0, g(r) = fp($PI) if r A, then r = fp($PI * k) for some k, and set g(r) = fp($PI * (k+1)) if r B, then g(r) = r; g maps [0,1] -> (0,1]. At least I think that works. /bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
On 11/30/06, Bernie Cosell <bernie@fantasyfarm.com> wrote:
As has been mentioned, there's no 'nice' way of doing it [in either direction, open->closed or vice versa].
Thanks Bernie, Franklin, and Dan for your responses. I suspected that it wasn't possible, or at least, not easy. Kerry -- lkmitch@gmail.com www.fractalus.com/kerry
participants (4)
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Bernie Cosell -
franktaw＠netscape.net -
Fred lunnon -
Kerry Mitchell