Continuous intervals for S(r) on both sides of integers and half-integers have been "known" since January, but not proved yet. I'm wrapping up a proof and can probably post it later this week. Examples: S(r) = r/5 on (2,15/7] S(r) = 1-3/r on [9/5,2) S(r) = r/4 on (3/2,8/5 S(r) = 1/3 on [4/3,3/2) (a special case) A new example from my recent posts is S(r) = (2r-1)/6 on (12/7,59/34] S is not continuous in a neighborhood of 1, however. - Scott
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Andy Latto Sent: Sunday, March 08, 2009 8:43 PM To: math-fun Subject: Re: [math-fun] muffin problem
Even if there were cases with S(km,kp) < S(m,p), the non-projective view doesn't encourage and might even preclude asking about and exploring the (many) cases where the projective view S(r) has continuous intervals.
We don't know there are any such intervals, do we? There is no open interval on which we know all the values of S, so we can't know that it is continuous anywhere.