[math-fun] Extremely discontinuous function
Intuitively, there ought to be a R->R function whose graph is dense in R^2. But I haven't been able to come up with one quickly. Is there a classic example?
Yes, also by the great late J. H. Conway: https://en.wikipedia.org/wiki/Conway_base_13_function It's actually stronger than what you've asked for; it takes every real value on every nontrivial interval. -- APG.
Sent: Tuesday, April 14, 2020 at 5:32 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Extremely discontinuous function
Intuitively, there ought to be a R->R function whose graph is dense in R^2. But I haven't been able to come up with one quickly. Is there a classic example? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_Every_ real value? Yes, that's much stronger! And obviously the previous mention of the base-13 function triggered my question, though I thought I hadn't heard of the function before -- the association must have been in my subconscious somewhere. All right, I'm going to have to read up on this miraculous function. (In case anybody was hoping that this topic might lead to interesting visuals -- well, no. I think we can prove that any presentation will just look like a gray blur.) On Tue, Apr 14, 2020 at 12:44 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
Yes, also by the great late J. H. Conway:
https://en.wikipedia.org/wiki/Conway_base_13_function
It's actually stronger than what you've asked for; it takes every real value on every nontrivial interval.
-- APG.
Sent: Tuesday, April 14, 2020 at 5:32 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Extremely discontinuous function
Intuitively, there ought to be a R->R function whose graph is dense in R^2. But I haven't been able to come up with one quickly. Is there a classic example? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Imre Leader set this problem [to find a function with this property that it takes every real value in every nontrivial interval] as an exercise. Raymond Wright's solution was a particularly elegant 'just do it' method: 1. Enumerate the sequence of nontrivial intervals with rational endpoints; let's call them I_1, I_2, I_3, ... 2. Let f_0 be the empty partial function (where everything is undefined); 3. After defining the partial function f_(n-1) whose support is nowhere dense by construction, we let J_n be a nontrivial subinterval of I_n such that f_(n-1) is not defined anywhere on J_n; 4. Take a Cantor set C_n contained in J_n, and take a surjection g_n from C_n to the reals R; 5. Let f_n be the partial function given by the (disjoint) union of the nowhere-densely-defined partial functions f_(n-1) and g_n; 6. Repeat steps {3,4,5} for each n in the naturals; 7. Let f_omega be the limit of (f_0, f_1, f_2, f_3, ...); 8. Define f(x) to be f_omega(x) if the latter is defined, and 0 otherwise.
Sent: Tuesday, April 14, 2020 at 5:50 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Extremely discontinuous function
_Every_ real value? Yes, that's much stronger! And obviously the previous mention of the base-13 function triggered my question, though I thought I hadn't heard of the function before -- the association must have been in my subconscious somewhere.
All right, I'm going to have to read up on this miraculous function.
(In case anybody was hoping that this topic might lead to interesting visuals -- well, no. I think we can prove that any presentation will just look like a gray blur.)
On Tue, Apr 14, 2020 at 12:44 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
Yes, also by the great late J. H. Conway:
https://en.wikipedia.org/wiki/Conway_base_13_function
It's actually stronger than what you've asked for; it takes every real value on every nontrivial interval.
-- APG.
Sent: Tuesday, April 14, 2020 at 5:32 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Extremely discontinuous function
Intuitively, there ought to be a R->R function whose graph is dense in R^2. But I haven't been able to come up with one quickly. Is there a classic example? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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A variation (arguably simpler) on the base-13 theme; map each real number to the supremum density of 1's in the binary representation of its fractional part. This maps every nonempty interval to all of [0,1]. Then use something like the tangent to stretch [0,1] to all of R (it does not matter what we do with the endpoints). On Tue, Apr 14, 2020 at 11:36 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
Imre Leader set this problem [to find a function with this property that it takes every real value in every nontrivial interval] as an exercise. Raymond Wright's solution was a particularly elegant 'just do it' method:
1. Enumerate the sequence of nontrivial intervals with rational endpoints; let's call them I_1, I_2, I_3, ...
2. Let f_0 be the empty partial function (where everything is undefined);
3. After defining the partial function f_(n-1) whose support is nowhere dense by construction, we let J_n be a nontrivial subinterval of I_n such that f_(n-1) is not defined anywhere on J_n;
4. Take a Cantor set C_n contained in J_n, and take a surjection g_n from C_n to the reals R;
5. Let f_n be the partial function given by the (disjoint) union of the nowhere-densely-defined partial functions f_(n-1) and g_n;
6. Repeat steps {3,4,5} for each n in the naturals;
7. Let f_omega be the limit of (f_0, f_1, f_2, f_3, ...);
8. Define f(x) to be f_omega(x) if the latter is defined, and 0 otherwise.
Sent: Tuesday, April 14, 2020 at 5:50 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Extremely discontinuous function
_Every_ real value? Yes, that's much stronger! And obviously the previous mention of the base-13 function triggered my question, though I thought I hadn't heard of the function before -- the association must have been in my subconscious somewhere.
All right, I'm going to have to read up on this miraculous function.
(In case anybody was hoping that this topic might lead to interesting visuals -- well, no. I think we can prove that any presentation will just look like a gray blur.)
On Tue, Apr 14, 2020 at 12:44 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
Yes, also by the great late J. H. Conway:
https://en.wikipedia.org/wiki/Conway_base_13_function
It's actually stronger than what you've asked for; it takes every real value on every nontrivial interval.
-- APG.
Sent: Tuesday, April 14, 2020 at 5:32 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Extremely discontinuous function
Intuitively, there ought to be a R->R function whose graph is dense in R^2. But I haven't been able to come up with one quickly. Is there a classic example? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On 14 Apr 2020 at 18:44, Adam P. Goucher wrote: Yes, also by the great late J. H. Conway:
https://en.wikipedia.org/wiki/Conway_base_13_function
It's actually stronger than what you've asked for; it takes every real value on every nontrivial interval.
how cool -- I came up with a proof using a similar idea, back as an undergraduate: the question was to prove that the algebraic numbers were countable.. I did it in base 13. with the digits 0123456789 + - ^ . With the convention that ^^N denotes the Nth solution to the polynomial. It was then , as with Conays' base 13 function it was easy to show that every algebraic number maps to an integer and so is countable. I did that a lotta years ago [as a sophomore] but I think it was a valid proof. I can't even remember what the "proper" proof was any more /Bernie\ Bernie Cosell bernie@fantasyfarm.com -- Too many people; too few sheep --
My first thought was that any discontinuous linear function has a graph that's dense in R2, but since their construction requires the axiom of choice, that's probably not what you're looking for But you can use the same construction, just stopping after countably many steps, to get the same effect. In fact, I think just one step suffices. Define f as If x = a + b sqrt(2), a, b, rational, then f(x) = a + 2b. otherwise, f(x) = 0. finding a, b, where f(x) ~ y means finding a, b where a + b sqrt (2) ~ x, and a+ 2b ~ y Solving these two linear equations gives the desired values of a and b. They won't be rational, but choosing a' and b' arbitrarily close to a and b will give a point in the graph of the function arbitrarily close to (x, y). Andy On Tue, Apr 14, 2020 at 12:32 PM Allan Wechsler <acwacw@gmail.com> wrote:
Intuitively, there ought to be a R->R function whose graph is dense in R^2. But I haven't been able to come up with one quickly. Is there a classic example? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
Andy Latto has given the sort of answer I was racking my brain for, though of course the "Base-13" function is much more of a tour-de-force. On Tue, Apr 14, 2020 at 12:57 PM Andy Latto <andy.latto@pobox.com> wrote:
My first thought was that any discontinuous linear function has a graph that's dense in R2, but since their construction requires the axiom of choice, that's probably not what you're looking for
But you can use the same construction, just stopping after countably many steps, to get the same effect. In fact, I think just one step suffices.
Define f as
If x = a + b sqrt(2), a, b, rational, then f(x) = a + 2b. otherwise, f(x) = 0.
finding a, b, where f(x) ~ y means finding a, b where a + b sqrt (2) ~ x, and a+ 2b ~ y
Solving these two linear equations gives the desired values of a and b. They won't be rational, but choosing a' and b' arbitrarily close to a and b will give a point in the graph of the function arbitrarily close to (x, y).
Andy
On Tue, Apr 14, 2020 at 12:32 PM Allan Wechsler <acwacw@gmail.com> wrote:
Intuitively, there ought to be a R->R function whose graph is dense in
R^2.
But I haven't been able to come up with one quickly. Is there a classic example? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
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participants (5)
-
Adam P. Goucher -
Allan Wechsler -
Andy Latto -
Bernie Cosell -
Michael Collins