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Measure zero and differentiability

by quasar987
Tags: differentiability, measure, solved
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quasar987
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Nov24-07, 10:40 AM
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1. The problem statement, all variables and given/known data
I proved in the preceding exercise the following characterization of measure zero:

"A subset E of R is of measure zero if and only if it has the following property:

(***) There exists a sequence [tex]J_k=]a_k,b_k[[/tex] such that every x in E belongs to an infinity of J_k and

[tex]\sum_{k=1}^{+\infty}(b_k-a_k)<+\infty[/tex]"

Now the question is the following:

Let E be of null measure and {J_k} be as above. Let also [tex]f_E:\mathbb{R}\rightarrow\mathbb{R}[/tex] be the increasing function defined by

[tex]f_E(x) = \sum_{k=1}^{+\infty}\lambda(]-\infty,x]\cap J_k)[/tex]

Show f is not differentiable at any point of E.


2. Relevant equations

Differentiable iff the limit of the differential quotient exists and is bounded iff the left and right derivative are equal

3. The attempt at a solution

Let h>0 and x0 be in E. I can write the right derivative and use the additivity of measure to simplify to

[tex]D_rf_E(x_0)=\lim_{h\rightarrow 0}\frac{\sum_{k=1}^{+\infty}\lambda([x,x+h]\cap J_k)}{h}[/tex]

then what? What difference does it make than x is in E? I mean, how does the fact that is belongs to an infinity of J_k comes in?
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EnumaElish
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Nov25-07, 09:50 PM
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Is "x + h" in J_k for some h > 0?
quasar987
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Nov25-07, 10:09 PM
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Well, each J_k is open and contains x, so for every k, there is an h, call it h_k such that x + h is in J_k, yes.

quasar987
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Nov27-07, 10:24 AM
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Measure zero and differentiability

Help!


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