On 2013-04-24, email@example.com <firstname.lastname@example.org> wrote: > On Wed, 24 Apr 2013 17:52:09 +0000 (UTC), Herman Rubin ><email@example.com> wrote:
>>On 2013-04-24, Dan <firstname.lastname@example.org> wrote: >>> On Apr 24, 5:38 pm, dullr...@sprynet.com wrote: >>>> On Wed, 24 Apr 2013 02:19:43 -0700, William Elliot <ma...@panix.com> >>>> wrote:
>>>> >Assume for f:[0,1] -> R that there's some c /= 0,1 with >>>> >for all x in [0,1/2], f(x) = c.f(2x).
>>>> >Show there's some k with for all x in [0,1], f(x) = kx.
>>For one thing, the theorem is false. The most one can hope >>to get is that f(x)=kx^q for some k and q>0. On the open >>interval, q can even be negative.
>>>> Not true. It's well known that there exists a nowhere- >>>> continuous function f : R -> R such that >>>> f(x+y) = f(x) + f(y) for all x, y.
>>>> Maybe you omitted a hypothesis? Seems like it may be >>>> true for continuous f...
>>One does not need to go that far. The general solution is >>f(x)=x^q*h(log_2(x)), where h is any function.
> ??? Since log_2 is 1-1, allowing h to be any function > here says that f can be any function.
It should be h is a periodic function of period 1.
Hardly. The condition specified requires that for all x, f(x), f(2x), and f(4x) are in geometric progression. But f can be any function satisfying this; the behavior between two arguments, one double the other, can be arbitrary given c, but f is dertermined completely from this.
>>Continuous >>does not get farther. However, all derivatives bounded >>will get q to be a positive integer and h constant.
>>> Maybe if you restrict it such that f:[0,1] -> [0,1]? >>> Just a hunch,not sure.
>>It can be seen from the above that this will not suffice. >>Even requiring that f be 1-1 will not suffice to get more >>than the general solution with limits on h, but not just >>the multiple of a power.
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University email@example.com Phone: (765)494-6054 FAX: (765)494-0558