Characterization of functions having anti-derivatives

[Grothendieck-Hirzebruch]

Can functions f: \R \to \R which have an anti-derivative be
characterized in some meaningful way? Clearly, such an f must have the
intermediate value property, but this is not sufficient. What about
the property that the set of discontinuity points is of the first
category?

[Valeriu Anisiu]

The function f(x) = sin(1/x) for x =/= 0 and f(0) = a
has the intermediate value property for a in [-1,1],
is discontinuous only at 0, but has an anti-derivative
only for a=0.

V. Anisiu

[Dave L. Renfro]

Grothendieck-Hirzebruch wrote:

The discontinuity sets for derivatives are the
F_sigma first category sets, but this collection
also characterizes the discontinuity sets for
semicontinuous functions (see [1]).

Every derivative is both Baire one (a pointwise
limit of continuous functions) and Darboux (has
the intermediate value property). Each of these
properties is nice in some ways, but when taken
together they provide us with something that is
much nicer than we might expect, kind of like
how "compact" and "Hausdorff" behave in general
topology, which when taken together form a class
of spaces far better behaved than you'd guess
based on how nice each of these individual
properties is. However, there exist Darboux
Baire one functions that are not derivatives.
Indeed, there are a number of Baire category
results that show "most" Darboux Baire one
functions are very far from being derivatives
(see [2]).

Derivatives also satisfy stronger versions
of the intermediate value property. For
example, f' having the intermediate value
property implies that the set E(a,b) =
{x in J: a < f'(x) < b} is either empty
or has cardinality of the continuum. Denjoy
(1916) (later rediscovered by J. A. Clarkson
in 1947) improved this by showing that each of
the sets E(a,b) is either empty or has positive
measure. A further strengthening of the IVP for
derivatives, stronger than this positive measure
result, was made by Zygmunt Zahorski [3]. Then
Clifford E. Weil [4] came up with a property
satisfied by all derivatives that is strictly
stronger than Zahorski's property and, after this,
Peter S. Bullen and D. N. Sarkhel [5] came up with
a property satisfied by all derivatives that
is strictly stronger than Weil's property. (This
last paper might not the strongest result so
far published, by the way.)

I believe William H. Young was the first to point
out how nice Darboux Baire one functions are,
although Young's work on this topic (done around 1907
to 1915) was largely unknown until the late 1950s
and early 1960s, when Solomon Marcus (Romanian),
Jan S. Lipinski (Polish), and a few other Eastern
European mathematicians began extending the work
that Young and some others did (Arnaud Denjoy-1910s,
Gustave Choquet-1947, some of Lusin's students-1920s
& 1930s, etc.). Andrew M. Bruckner began reading
the work of these Eastern European mathematicians
in the early 1960s and, throughout his career,
wrote many papers about (and directed student
Ph.D. Dissertations on) the relationship between
the class of derivatives and the class of Darboux
Baire one functions. Some of Bruckner's several
expository papers on this topic, the survey paper
he and his first Ph.D. student wrote [6], and his
book [7] are where you should begin if you want
to look into this subject.

The problem of finding some kind of descriptive
or topological way to classify derivatives (such as
classifying/describing the inverse images of open
intervals under derivatives) was the driving force
behind much of this work, a problem that I think
originates from William H. Young. Relatively
recently (1990s), some logicians got involved and
showed that, in a certain sense, no relatively
simple characterization is possible [8].

[1]
http://groups.google.com/group/sci.math/msg/05dbc0ee4c69898e

[2] Jack Ceder and T. L. Pearson, "On typical bounded
darboux baire one functions", Acta Mathematica
Hungarica 37 (1981), 339-348.
I. Mustafa, "On residual subsets of Darboux Baire
class 1 functions, Real Analysis Exchange 9
(1983-84), 394-395.
Michael J. Evans and Paul D. Humke, "A typical
property of Baire 1 Darboux functions", Proceedings
of the American Mathematical Society 98 (1986), 441-447

[3] Zygmunt Zahorski, "Sur la primiere derivee", Transactions
of the American Mathematical Society 69 (1950), 1-54.

[4] Clifford E. Weil, "A property for certain derivatives",
Indiana University Mathematics Journal 23 (1973-74),
527-536.

[5] Peter S. Bullen and D. N. Sarkhel, "A continuity-like
property of derivatives", Canadian Mathematical
Bulletin 39 (1996), 10-20.
http://books.google.com/books?id=f__76i50BKIC&q=Zahorski

[6] Andrew M. Bruckner and John L. Leonard, "Derivatives",
American Mathematical Monthly 73 #4 (April 1966)
[Part II: Papers in Analysis, Herbert Ellsworth
Slaught Memorial Papers #11], 24-56.

[7] Andrew M. Bruckner, "Differentiation of Real Functions",
2'nd edition, CRM Monograph Series #5, American
Mathematical Society, 1994, 195 pages. [The second
edition is essentially unchanged from the first
edition (Lecture Notes in Mathematics #659,
Springer-Verlag, 1978), with the exception of
a new chapter on recent developments.]

[8]
http://www.google.com/search?q=complexity-of-antidifferentiation

Dave L. Renfro