Ponder This Challenge - October 2001 - f maps any open interval onto R

Ponder This Challenge:

This month's puzzle comes from Laszi Ladanyi, who heard it in college.

Find a function f from R (that is, the set of real numbers) to R
such that for all open intervals I=(u,v) ,we have that f maps (u,v) onto R.
The function f is independent of u,v.

(That is, given any target real number t, and any interval
I=(u,v) with u<v, there is x with u<x<v and f(x)=t.)

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If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

• Click here to view the solution

  First define a function g from R to the half-open interval [0,1).

  Express |x| in ternary (base 3).
  (Note: when x is rational with a power of 3 in the denominator, we have a choice between a representation ending in an infinite number of 0's or one ending in an infinite number of 2's; select the former.)

  If the expression has an infinite number of 2's, or no 2's at all, then set g(x)=0.
  Otherwise it has a finite number of 2's.

  Take the portion of the ternary expansion of |x| after the last 2, precede it with a decimal point, and interpret it as a binary number; let that be g(x).
  The function g maps R to [0,1), and it maps each open interval I=(u,v) to all of [0,1).

  Now set f(x)=0 if g(x)=0, otherwise f(x)=tan(pi(x-1/2)).

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  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Bill Roscoe (10.01.2001 @ 11:35 AM EDT)
• Henry Bottomley (10.01.2001 @ 12:22 PM EDT)
• Vineet Gupta (10.01.2001 @ 2:56 PM EDT)
• George Ross (10.02.2001 @ 2:35 PM EDT)
• Alexey Vorobyov (10.02.2001 @ 2:51 PM EDT)
• Ionel Santa (10.03.2001 @ 2:39 AM EDT)
• Jacques Willekens (10.03.2001 @ 3:13 AM EDT)
• Vadim Alexandrov (10.03.2001 @ 8:40 AM EDT)
• James McGrath (10.03.2001 @ 2:40 PM EDT)
• Marian Olteanu (10.04.2001 @ 3:52 PM EDT)
• Ruth Algor (10.03.2001 @ 8:46 PM EDT)
• Andreas Spillner (10.04.2001 @ 4:07 AM EDT)
• Hilary Jones (10.04.2001 @ 9:54 PM EDT)
• Ben North (10.05.2001 @ 10:49 AM EDT)
• Andy Lowry & Howie Kaye (10.05.2001 @ 11:44 AM EDT)
• Vladimir Sedach (10.05.2001 @ 3:35 PM EDT)
• Sameer Udeshi (10.05.2001 @ 3:52 PM EDT)
• Saibal Mitra (10.06.2001 @ 6:26 PM EDT)
• Marian Vasile (10.08.2001 @ 10:31 AM EDT)
• Felipe Bertrand and Cesar Sanchez (10.09.2001 @ 2:08 PM EDT)
• Oleg Mazurov (10.10.2001 @ 1:26 AM EDT)
• Jens Voss (10.10.2001 @ 3:47 AM EDT)
• David G Radcliffe (10.12.2001 @ 4:05 PM EDT)
• Chi-Bong Chan (10.13.2001 @ 3:39 PM EDT)
• Ross Millikan (10.15.2001 @ 11:26 AM EDT)
• G XU (10.15.2001 @ 1:15 PM EDT)
• Victor Chang (10.15.2001 @ 8:22 PM EDT)
• Koen Claessen (10.16.2001 @ 4:49 AM EDT)
• Christopher Howe (10.17.2001 @ 7:04 PM EDT)
• Vishaal Kapoor (10.20.2001 @ 3:45 AM EDT)
• John Lamping (10.20.2001 @ 3:04 PM EDT)
• Devin Sezer (10.26.2001 @ 8:37 PM EDT)