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Re: Discontinuity of a derivative function
Posted:
Dec 11, 2000 5:08 AM


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 nondense 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={xf 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]) = ba 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, 163179]
[Z] Zahorski, Z. Sur la premiÃÂÃÂ¨re dÃÂÃÂ©rivÃÂÃÂ©e. Trans. Amer. Math. Soc. 69, (1950). 154.
Best regards Lars



