Which functions are derivatives of some other function?

Question

There is a fundamental formula in integral calculus which states that $\int_a^bF'(x)dx=F(b)-F(a)$. This formula gives a connection between definite and indefinite integrals. There are plenty of functions which are integrable (in the definite sense)-for example each bounded measurable function is ok. However I don't know precisely which functions do have indefinite integral: in other words which functions are derivatives of some other function? For example continuity is enough: on the other hand one cas show that each function being the derivative (although need not to be continuous) has the Darboux property. So my question is

Which functions are derivatives of some other function?

Answer 1

The problem is a rather famous (and natural) one. It was first explicitly stated by W. H. Young in 1911. I like quoting this passage so much (even if only to find an excuse to use the word "mooted") that I won't resist that impulse here.

"Recent research [of Lebesgue and Vitali] has provided us with a set of necessary and sufficient conditions that a function may be the indefinite integral..., 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. The necessary conditions in question are of considerable importance and interest. A function which is a differential coefficient has, in fact, various striking properties. It must be pointwise discontinuous with respect to every perfect set; it can have no discontinuities of the first kind; it assumes in every interval all values between its upper and lower bounds in that interval; its value at every point is one of the limits on both sides of the values in the neighbourhood; its upper and lower bounds, when finite, are unaltered, if we omit the values at any countable set of points; the points at which it is infinite form an inner limiting set of content zero. From these necessary conditions we are able to deduce much valuable information as to when a function is certainly not a differential coefficient . . . . These conditions do not, however, render us any material assistance, even in answering the simple question as to whether the product of two differential coefficients is a differential coefficient, and this not even in the special case in which one of the differential coefficients is a continuous function." ...from W H Young, A note on the property of being a differential coefficient. Proc. London Math. Soc. 1911 (2) 9, 360-368.

In the monograph by Andrew M. Bruckner, Differentiation of real functions, Chapter seven there is a discussion of the problem of characterizing derivatives.

Andy has updated his account of this problem in a survey article for the Real Analysis Exchange:

Bruckner, Andrew M. The problem of characterizing derivatives revisited. Real Anal. Exchange 21 (1995/96), no. 1, 112--133.

Here are a few actual characterizations of derivatives:

C. Neugebauer, Darboux functions of Baire class 1 and derivatives, Proc. Amer. Math. Soc., 13 (1962), 838–843.

D. Preiss and M. Tartaglia, On Characterizing Derivatives, Proceedings of the American Mathematical Society, Vol. 123, No. 8 (Aug., 1995), 2417-2420.

Chris Freiling, On the problem of characterizing derivatives, Real Analysis Exchange 23 (1997/98), no. 2, 805-812.

Brian S. Thomson, On Riemann Sums, Real Analysis Exchange 37 (2011/12), 1-22.

My guess is that you won't find any of these satisfying in the way, for example, that "continuous" functions have multiple necessary and sufficient characterizations, most of them natural and compelling.

Answer 2

If $f(x)$ is any continuous function on the interval $[a,b]$, then for any $c\in [a,b]$ we can define the function $F(x) = \int_c^xf(t)dt$ for $x\in(a,b)$. In this way, the Fundamental Theorem of Calculus says that $F'(x) = f(x)$. So any continuous function $f(x)$ is a derivative of some other function and $\int_c^xf(t)dt$ is an antiderivative.

While "all continuous functions" is a large class, continuity is not necessary for a function to be a derivative. This Post gives many examples of some fairly weird functions whose derivatives are not continuous.