Thursday, December 27, 2012

The problem of characterizing derivatives

I noticed that someone had asked a question not too long ago on the MathOverFlow site about characterizing derivatives.  The answers were somewhat helpful but none of the usual experts seems to have dived into the discussion, so there was only limited information. 

I did add a comment, but maybe I should post some stuff here for those who happen to have some interest in the problem.

The problem naively is to find some necessary and sufficient condition in order that a function  f: [a,b] --> R  would be the derivative everywhere in [a,b] of some other function.   (For example continuity would be sufficient but certainly not necessary,  Baire class 1 would be necessary but not sufficient.)

To research the problem you should consult at least the following:

Andrew M. Bruckner, Differentiation of real functions. Lecture Notes in Math., vol. 659, Springer-Verlag, Berlin and New York, 1978, x + 246 pp.
or the more recent edition:
Differentiation of real functions.
Second edition. CRM Monograph Series 5.  American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6 

Chapter seven contains an entertaining and accessible account of the problem, including the original formulation of the problem by W. H. Young.

Andy has updated his discussion of this problem in a 1995 survey article written for the Real Analysis Exchange:   

Bruckner, Andrew M.  The problem of characterizing derivatives revisited.  Real Anal. Exchange  21  (1995/96),  no. 1, 112--133.
 Since then there have been a few varied expressions of conditions that characterize derivatives in perhaps unusual ways:
  1. On Characterizing Derivatives,  D. Preiss and M. Tartaglia Proceedings of the American Mathematical Society, Vol. 123, No. 8 (Aug., 1995), pp. 2417-2420
  2. Freiling, Chris,   On the problem of characterizing derivatives.
    Real Anal. Exchange 23 (1997/98), no. 2, 805–812.
  3. Brian S. Thomson, On Riemann Sums, Real Analysis Exchange Vol. 37(1), 2011/2012, pp. 1–22

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