## Neugebauer’s theorem

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 $f$ such that $f(x)=\sin(1/x)$ for $x\ne 0$, and $f(0)\in[-1,1]$ is Darboux continuous and Baire one, but only the function with $f(0)=0$ 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 $f:[0,1]\to\mathbb R$, although this makes no real difference. Whenever an interval is mentioned, it is understood to be nondegenerate. For any closed subinterval $I\subseteq [0,1]$, we write $I^\circ$ for its interior, and $\mathrm{lh}(I)$ for its length. Given a point $x\in[0,1]$, we write $I\to x$ to indicate that $\mathrm{lh}(I)\to 0$ and $x\in I$.

Theorem (Neugebauer). A function $f:[0,1]\to\mathbb R$ is a derivative iff to each closed subinterval $I$ of ${}[0,1]$ we can associate a point $x_I\in I^\circ$ in such a way that the following hold:

1. For all $x\in[0,1]$, if $I\to x$, then $f(x_I)\to f(x)$, and

2. For all  closed subintervals $I,I_1,I_2$ of ${}[0,1]$, if $I=I_1\cup I_2$ and ${I_1}^\circ\cap {I_2}^\circ=\emptyset$, then $\mathrm{lh}(I)f(x_I)=\mathrm{lh}(I_1)f(x_{I_1})+\mathrm{lh}(I_2)f(x_{I_2})$.

The proof is remarkably simple.

Proof. Assume first that $f$ is a derivative, say $f=F'$. By the mean value theorem, for each interval $I=[a,b]\subseteq[0,1]$ we can find at least one point $c\in(a,b)$ such that $\displaystyle \frac{F(b)-F(a)}{b-a}=f(c)$. Pick any such $c$ and call it $x_I$. Condition 2 above is immediate.

To see that condition 1 also holds, fix $x\in[0,1]$, and note that for $h$ close enough to $0$ with $x+h\in [0,1]$, we have that

$F(x+h)=F(x)+hf(x)+O(h^2)$.

It follows that if $h,j\to0^+$, then

$\displaystyle \frac{F(x+h)-F(x-j)}{(x+h)-(x-j)}=$ $\displaystyle \frac{(F(x+h)-F(x))+(F(x)-F(x-j))}{h+j}=$ $\displaystyle \frac{(h+j)f(x)+O(h^2)+O(j^2)}{h+j}= f(x)+\frac{O((h+j)^2)}{h+j}\to f(x)$,

that is, if $I\to x$, then $f(x_I)\to f(x)$.

Conversely, suppose that conditions 1 and 2 hold. For each closed interval $I\subseteq[0,1]$, let $x_I\in I^\circ$ be as required. Define $F$ so that

$F(0)=0$

and, for $0, let $I=[0,x]$, and set

$F(x)=f(x_I)\mathrm{lh}(I)=xf(x_I)$.

One easily verifies (from condition 2) that if $0\le a and $I=[a,b]$, then $F(b)-F(a)=(b-a)f(x_I)$. It follows from condition 1 that

$\displaystyle \frac{F(x+h)-F(x)}{h}\to f(x)$

as $h\to 0$, that is, $f=F'$. This completes the proof. $\Box$

In the same paper, Neugebauer also shows that if $f:[0,1]\to\mathbb R$ is a Darboux continuous Baire one function, then to each closed subinterval $I\subseteq [0,1]$ we can associate a point $x_I\in I^\circ$ 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 $F$ with $F'=f$, for each $I=[a,b]$ we need to pick an $x_I\in(a,b)$ with $\displaystyle {F(b)-F(a)}{b-a}=f(x_I)$. 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 $a, if $c=(F(b)-F(a))/(b-a)$, the set $\{x\in(a,b)\mid f(x)=c\}$ is Borel (in fact, $G_\delta$). The result follows (albeit with a $\mathbf{\Pi}^1_1$ 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 $G_\delta$ 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 $C^*$-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 $F:[a,b]\to\mathbb R$ that is differentiable in $(a,b)$, define $g(x)=F(x)-y(x)$, where $y$ is the line that goes through the points $(a,F(a))$ and $(b,F(b))$. We see that $g'(x)=0$ iff $\displaystyle F'(x)=\frac{F(b)-F(a)}{b-a}$, and that $g(a)=g(b)=0$.

It follows that we are done if, given a continuous $g:[a,b]\to\mathbb R$ that is differentiable in $(a,b)$ and satisfies $g(a)=g(b)=0$, we can definably pick a point $c\in(a,b)$ with $g'(c)=0$. The following procedure attains this:

• Let $c$ be the supremum of the set of points $x\in(a,b)$ where $g$ attains its (global) maximum, provided that this set is non-empty. If this is the case, note that (by continuity), the function $g$ also attains its maximum at the point $c$ and therefore, if $c, then $g'(c)=0$. This works, unless the global maximum of $g$ is attained at $a$ and $b$. In that case:
• Let $c$ be the supremum of the set of points $x\in(a,b)$ where $g$ attains its (global) minimum, provided that this set is non-empty. This works, unless $g$ also attains its minimum at $a$ and $b$. But this means that $g$ is constant (identically equal to $0$ ). In that case:
• Let $c=(a+b)/2$.

(Essentially, what we are doing here is to replace the $G_\delta$ 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.]