Neugebauer’s theorem

December 30, 2014

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 $\mathrm{lh}(I)f(x_I)=\mathrm{lh}(I_1)f(x_{I_1})+\mathrm{lh}(I_2)f(x_{I_2})$.