Re: local boundedness of derivative

Dusan wrote:

Consider a function f: (a,b)-->R which has a finite derivative f '
on (a,b). I wonder if the derivative f ' must be locally bounded
on (a,b), i.e. for each x\in (a,b) there exists a neighbourhood U
of x such that f ' is bounded on U.

Jules has pointed out that (x^2)*sin(1 / x^2) is not locally
bounded at x=0. The existence of this behavior was a limitation
of Lebesgue integration for inverting (everywhere finite)
derivatives that led to the Denjoy, Khintchine, and Perron
integrals during the period from 1912 to 1918. See Gordon [1]
(beginning of Chapters 7 & 8, pp. 107-108 & 121) and Pesin [3]
(Chapters 8 & 9). To be more precise, the issues were with
the local non-summability (i.e. non-integrability) of the
derivative at various points, not the weaker notion of local
non-boundedness. However, one can show that the exceptional
set of local non-summability has the same "size" and "structure"
as the set of local non-boundedness, namely a set which is
both closed and nowhere dense. (Of course, you can have functions
where these sets are different. I'm just saying that both sets
are characterized by these two properties.)

To begin with, the set of local non-boundedness of _any_ function
has to be a closed set. (It's easy to show that the complement
is open.)

Let f' be finite everywhere. Then f' is a Baire one function,
and hence its points of continuity are dense in the real line.
It is not difficult to see that this implies the set of local
non-boundedness of f' must be nowhere dense. Therefore, the set
of local non-boundedness of f' is closed and nowhere dense.

Let C be any closed and nowhere dense set on the real line.
Using a Volterra type construction with (x^2)*sin(1 / x^2),
we can construct a function f: R --> R such that f' is finite
at each point in the real line and f' is not locally bounded
at each point in C. See Jeffery [2] (Section 6.2, Example 6.2,
pp. 148-149) for the details.

[1] Russell A. Gordon, "The Integrals of Lebesgue, Denjoy,
Perron, and Henstock", Graduate Studies in Mathematics #4,
The American Mathematical Society, 1994, xii + 395 pages.

[2] Ralph L. Jeffery, "The Theory of Functions of a Real
Variable", Dover Publications, 1953/1985, xiv + 232 pages.

[3] Ivan N. Pesin, "Classical and Modern Integration Theories",
translated and edited by Samuel Kotz, (Series) Probability
and Mathematical Statistics, Academic Press, 1970,
xx + 195 pages.

Dave L. Renfro

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