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Topic: Discontinuity of a derivative function
Replies: 5   Last Post: Dec 11, 2000 5:08 AM

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 Lars Olsen Posts: 18 Registered: 12/13/04
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 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 premiÃÂÃÂ¨re dÃÂÃÂ©rivÃÂÃÂ©e. Trans. Amer. Math. Soc. 69,
(1950). 1--54.

Best regards Lars

Date Subject Author
12/9/00 Nir
12/9/00 Paolo
12/9/00 Peter Ammon
12/9/00 G. A. Edgar
12/10/00 David C. Ullrich
12/11/00 Lars Olsen