Re: absolut continuity...?

On 24 May 2005 19:53:16 -0700, alexandru.lupas@xxxxxxxxxxx wrote:

>Find a function f:[a,b]--->R having the properties
>
> i) f is absolutely continuous on [a,b] ,
>
> ii) equation f(x)=0, x in [a,b], has at least
>
> one solution,
>
> iii) if p in (0,1), then function F:[a,b]---> R, F(x)=|f(x)|^p ,
>
> is'nt absolutely continuous on [a,b].

Start with a sequence c_n > 0 such that sum c_n < infinity
but sum c_n^p = infinity for all p in (0,1), for example,
c_n = ___. Choose x_1 < y_1 < x_2 < y_2 ... in [a,b] and
consider a piecewise-linear f with f(x_n) = 0 and
f(y_n) = c_n.


************************

David C. Ullrich
.

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