The problem of characterizing when a given function is a derivative has been studied at least since 1911 when Young published a paper recognizing its relevance:
W. H. Young. A note on the property of being a differential coefficient, Proc. Lond. Math. Soc., 9 (2), (1911), 360–368.
The paper opens with the following comment (Young refers to derivatives as “differential coefficients”):
Recent research has provided us with a set of necessary and sufficient conditions that a function may be an indefinite integral, in the generalised sense, of another function, and the way has thus been opened to important developments. The corresponding, much more difficult, problem of determining necessary and sufficient conditions that a function may be a differential coefficient, has barely been mooted; indeed, though we know a number of necessary conditions, no set even of sufficient conditions has to my knowledge ever been formulated, except that involved in the obvious statement that a continuous function is a differential coefficient.
It is more or less understood nowadays that no completely satisfactory characterization is possible. We know that derivatives are Darboux continuous (that is, they satisfy the intermediate value property), and are Baire one functions (that is, they are the pointwise limit of a sequence of continuous functions). But this is not enough: For instance, any function such that
for
, and
is Darboux continuous and Baire one, but only the function with
is a derivative.
Andrew Bruckner has written excellent surveys on derivatives and the characterization problem. See for instance:
Andrew M. Bruckner and John L. Leonard. Derivatives. Amer. Math. Monthly, 73 (4, part II), (1966), 24–56. MR0197632 (33 #5797).
Andrew M. Bruckner. Differentiation of real functions. Second edition. CRM Monograph Series, 5. American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6. MR1274044 (94m:26001).
Andrew M. Bruckner. The problem of characterizing derivatives revisited. Real Anal. Exchange, 21 (1), (1995/96), 112–133. MR1377522 (97g:26004).
Here I want to discuss briefly a characterization obtained by Neugebauer, see
Christoph Johannes Neugebauer. Darboux functions of Baire class one and derivatives. Proc. Amer. Math. Soc., 13 (6), (1962), 838–843. MR0143850 (26 #1400).
For concreteness, I will restrict discussion to functions , although this makes no real difference. Whenever an interval is mentioned, it is understood to be nondegenerate. For any closed subinterval
, we write
for its interior, and
for its length. Given a point
, we write
to indicate that
and
.
Theorem (Neugebauer). A function
is a derivative iff to each closed subinterval
of
we can associate a point
in such a way that the following hold:
-
For all
, if
, then
, and
-
For all closed subintervals
of
, if
and
, then
.
The proof is remarkably simple.
Proof. Assume first that is a derivative, say
. By the mean value theorem, for each interval
we can find at least one point
such that
. Pick any such
and call it
. Condition 2 above is immediate.
To see that condition 1 also holds, fix , and note that for
close enough to
with
, we have that
.
It follows that if , then
,
that is, if , then
.
Conversely, suppose that conditions 1 and 2 hold. For each closed interval , let
be as required. Define
so that
and, for , let
, and set
.
One easily verifies (from condition 2) that if and
, then
. It follows from condition 1 that
as , that is,
. This completes the proof.
In the same paper, Neugebauer also shows that if is a Darboux continuous Baire one function, then to each closed subinterval
we can associate a point
in such a way that condition 1 above holds. That is, it is precisely the possibility of strengthening condition 1 with condition 2 that distinguishes derivatives among the Darboux continuous Baire one functions.
A technical remark is in order. As written, the proof above seems to make blatant use of the axiom of choice: Given a differentiable function with
, for each
we need to pick an
with
. It seems useful to know that this can be done definably (indeed, in a Borel fashion) so that no appeal to choice is actually required.
There are two ways of verifying this. The first, general, approach, is to note that for any , if
, the set
is Borel (in fact,
). The result follows (albeit with a
function) from the Lusin-Sierpiński uniformization theorem. In fact, I suspect we can obtain a Borel uniformization here on general grounds, since the sections are
and “uniformly defined”. See for instance the appendix to:
George A. Elliott, Ilijas Farah, Vern Paulsen, Christian Rosendal, Andrew S. Toms, and Asger Törnquist. The isomorphism relation for separable
-algebras. Math. Res. Lett., 20 (6), (2013), 1071–1080. MR3228621.
The second, elementary, approach is specific to the situation at hand but does not require knowledge of descriptive set theory. It was indicated by TonyK on Math.Stackexchange, see here. The idea is to recall that the mean value theorem is proved as a corollary of Rolle’s theorem: Given a continuous that is differentiable in
, define
, where
is the line that goes through the points
and
. We see that
iff
, and that
.
It follows that we are done if, given a continuous that is differentiable in
and satisfies
, we can definably pick a point
with
. The following procedure attains this:
- Let
be the supremum of the set of points
where
attains its (global) maximum, provided that this set is non-empty. If this is the case, note that (by continuity), the function
also attains its maximum at the point
and therefore, if
, then
. This works, unless the global maximum of
is attained at
and
. In that case:
- Let
be the supremum of the set of points
where
attains its (global) minimum, provided that this set is non-empty. This works, unless
also attains its minimum at
and
. But this means that
is constant (identically equal to
). In that case:
- Let
.
(Essentially, what we are doing here is to replace the sections with closed subsets, in a uniform fashion.)
[It was in order to discuss these remarks that I talked recently at the Set Theory seminar here at Boise on the uniformization theorem. Thanks to Sam Coskey for his interest on this topic.]