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Topic: Ask-an-Analysis problem
Replies: 10   Last Post: Apr 25, 2013 1:01 PM

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Herman Rubin

Posts: 399
Registered: 2/4/10
Re: Ask-an-Analysis problem
Posted: Apr 25, 2013 1:01 PM
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On 2013-04-24, <> wrote:
> On Wed, 24 Apr 2013 17:52:09 +0000 (UTC), Herman Rubin
><> wrote:

>>On 2013-04-24, Dan <> wrote:
>>> On Apr 24, 5:38 pm, wrote:
>>>> On Wed, 24 Apr 2013 02:19:43 -0700, William Elliot <>
>>>> 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.

>>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 Phone: (765)494-6054 FAX: (765)494-0558

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