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Re: Discontinuity of a derivative function
Posted:
Dec 11, 2000 5:08 AM
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In article <3A32A113.B17B42C7@cornell.edu>, pa44@cornell.edu wrote:
> Nir wrote:
> >
> > Is there any restriction on the "quantity" of the points of
> > discontinuity of a function which is a derivative of another function
> > ? (For example, does the set of all points of discontinuity of a
> > derivative function have to be of Lebesgue measure zero or something
> > like that ?)
>
> Although I couldn't construct one, there are functions that are
> everywhere differentiable but whose derivative is nowhere continuous.
> So I'm guessing that there are no restrictions.
>
> -Peter
Not quite true.
For a function g:R\to R let Cont(g) denote the set of continuity points of g.
A function g:R\to R that is the _pointwise_ limit of a sequence of
continuous functions is called a Baire 1 function.
The following result is a classical result (going back to Baire), cf. [K,
(24.14)]
Theorem. If f:R\to R is Baire 1, then Cont(f) is dense in R (in fact,
Cont(f) is a comeager G_delta set).
Let E be any non-dense G_delta subset of R (e.g. E could be the empty
set), then (since a derivative is Baire 1) the above Theorem implies that
there is no differentiable function f:R\to R such that
Cont(f') = E.
In particular, there is no differentiable function f:R\to R such that
Cont(f') = the empty set.
In fact, the following more general result, due to Zahorski [Z], holds.
Theorem. Let M be a subset of R. Then the following are equivalent:
(1) There exists a continuous function f:R\to R such that
M={x|f is differentiable at x}
(2) There exist a G_delta set A and a F_sigma_delta set B with:
Lebesgue measure of (B intersection [a,b]) = b-a for all a<b
such that:
M = (A intersection B)
For a nice survey of these and related topics see [B].
References
[B] Bruckner, Andrew Differentiation of real functions. Second edition. CRM
Monograph Series, 5. American Mathematical Society, Providence, RI, 1994
[K] Kechris, Classical Descriptive Settheory, Springer Verlag.
[KL] Ki & Linton. Normal numbers and subsets of $\bold N$ with given densities.
Fund. Math. 144 (1994), no. 2, 163--179]
[Z] Zahorski, Z. Sur la premiÃÂère dÃÂérivÃÂée. Trans. Amer. Math. Soc. 69,
(1950). 1--54.
Best regards Lars
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