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