From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: Help: symmetric derivative Date: 3 Oct 1999 14:54:45 -0400 Newsgroups: sci.math G.E. Ivey [sci.math 1 Oct 99 20:34:18 -0400 (EDT)] wrote >Ben Logan said: >>Just one more question: according to my calculations, >>the symmetric derivative of abs(x) is 0. Now, the >>derivative of abs(x) exists for all x except 0 and the >>symmetric derivative is equal to it for all x except zero. >>So, just out of curiosity, wouldn't that mean that the >>symmetric derivative of the abs(x) is not a continuous function? > > Yep, that's what it means. Nothing at all unusual about >that. The "symmetric derivative" of |x| is 1 if x>0, 0 for >x= 0 and -1 if x<0. The regular derivative is also not >continuous at 0. The regular derivative of |x| is 1 >if x>0, -1 if x< 0 and does not exist for x=0. Elementary calculus texts aside (which do this for pedagogical reasons, I suspect), one typically says that the derivative of |x| has no continuous extension to x=0, rather than that it fails to have a derivative at x=0. There are a number of results known about the set of points where a function can be symmetrically differentiable, but not differentiable. Let I be an open interval and f : I --> reals. Denote by D(f) the set of points in I at which f has a finite derivative and denote by SD(f) the set of points in I at which f has a finite symmetric derivative. 1. As several others have observed, D(f) is a subset of SD(f). 2. If f is measurable, then the relative complement of D(f) in SD(f), SD(f) - D(f), has measure zero. A. Khintchine, "Recherches sur la structure des fonctions measurables", Fund. Math. 9 (1927), 212-279. 3. The continuum hypothesis implies that there exists a measurable function f such that SD(f) - D(f) is residual in I. [Residual in I means "has a first (Baire) category complement in I".] Let G be an additive subgroup of the reals that has second category in every interval [Erdos constructs such a group using the continuum hypothesis in "Some remarks on subgroups of real numbers", Colloq. Math. 42 (1979), 119-120.], and let f be the characteristic function of G. Then it is not difficult to check that SD(f) = G and D(f) = empty set. I don't know if there exists in ZFC a function f such that SD(f) - D(f) is residual. My guess would be "yes". (The answer may already be known.) 4. If the set of points at which f is continuous is dense in I, then SD(f) - D(f) is both first category and measure zero. In fact, the set is sigma-porous, a notion strictly stronger than "first category and measure zero". C. L. Belna, Michael J. Evans, and Paul D. Humke, "Symmetric and ordinary differentiation", Proc. Amer. Math. Soc. 72 (1978), 261-267. 5. There exist functions f continuous everywhere on I such that SD(f) - D(f) contains a nonempty perfect set. [In other words, even for everywhere continuous functions, SD(f) - D(f) can be uncountable.] James Foran, "The symmetric and ordinary derivative", Real Analysis Exchange 2 (1977), 105-108. Foran asks in this paper whether the set can have positive Hausdorff dimension. As far as I know, this is still open. 6. If the points at which f is continuous is dense in I, then SD(f) - D(f) is a countable union of uniformly symmetrically porous sets, where the uniformity constant for each set in this countable union can be uniformly chosen larger than any fixed number less than 1. It is known that such sets can still have Hausdorff dimension 1. Apparently, it is not known whether or not SD(f) - D(f) MUST have Hausdorff dimension less than 1, nor whether or not SD(f) - D(f) CAN have Hausdorff dimension greater than 0. (However, this may no longer be the case.) Ludek Zajicek, "A note on the symmetric and ordinary derivative", Atti Sem. Mat. Fis Univ. Modena 41 (1993), 263-267. [Proved the result for f continuous everywhere on I.] Michael J. Evans, "A note on symmetric and ordinary differentiation", Real Analysis Exchange 17 (1991-92), 820-826. [Generalized Zajicek's result for functions whose points of continuity are dense in I.] 7. The most complete reference available on these and other related matters is Brian S. Thomson, SYMMETRIC PROPERTIES OF REAL FUNCTIONS, Monographs and Textbooks in Pure and Applied Mathematics 183, Marcel Dekker, 1994, xiii + 447 pages. I don't have a copy of this book, but I feel certain that everything above (except possibly #8) can be found in it. Dave L. Renfro