
Re: References for Continuity Sets
Posted:
Dec 20, 2006 4:23 PM


This is a slightly expanded version of the version I posted on December 9. A more precise form of Theorem 2' is given and new references are [24], [26], [36], [45], [46], and [48].
This post is intended to archive some literature references for characterizations of the continuity sets of monotone, arbitrary, Baire one, and Riemann integrable functions. I did not include a reference for Theorem 1 unless I also cite that reference for at least one of the other theorems.
In what follows, "countable" means finite or countably infinite, and R denotes the set of real numbers with their usual metric and ordering properties.
We say a realvalued function f is Baire one if f is the pointwise limit of some sequence of continuous functions. Examples of Baire one functions are functions with countably many points of continuity, semicontinuous functions, derivatives, and functions f:R^2 > R that are separately continuous in each variable (but not necessarily functions from R^n to R for n > 2). A function is Baire two if it is a pointwise limit of Baire one functions.
Given a function f:R > R, we let C(f) and D(f) denote the sets of continuity and discontinuity points, respectively, of f.

THEOREM 1: If f:R > R is monotone (or even of bounded variation), then D(f) is countable.
THEOREM 1': If E is any countable subset of R, then there exists a strictly increasing function f:R > R such that D(f) = E.
THEOREM 2: If f:R > R is an arbitrary function, then D(f) is an F_sigma subset of R.
THEOREM 2': Given any F_sigma subset E of R, then there exists a Baire two function f:R > R such that D(f) = E.
REMARK: Very few, if any, of the references point out that f can be chosen to be a Baire two function, but the actual constructions are clearly Baire two functions.
THEOREM 3: If f:R > R is a Baire one function, then D(f) is an F_sigma meager (= first category) subset of R.
REMARK: This implies that, for each Baire one function, C(f) is dense in R, cdense in R, and even comeager in every open interval.
THEOREM 3': Given any F_sigma meager subset E of R, then there exists a Baire one function such that D(f) = E. In fact, f can be chosen to be semicontinuous or to be a bounded derivative.
REMARK: A meager F_sigma set can be cdense in R, have a measure zero complement, or to even have a Haudsorff hmeasure zero complement for any preassigned measure function h.
THEOREM 4: If f:[a,b] > R is Riemann integrable, then D(f) is an F_sigma meager & measure zero subset of R.
THEOREM 4': Given any F_sigma meager & measure zero subset E of [a,b], then there exists a Riemann integrable function f:[a,b] > R such that D(f) = E.

PROOFS OF 1:
Boa [4] (Section 22, bottom of p. 159) BBT [7] (Exercise 1:3.14 gives a method to be verified) BBT [8] (Theorem 5.60, p. 247) B/K [10] (Chapter 131, Corollary 2, p. 275) Car [11] (Chapter 2, Theorem 2.17, p. 32) For [15] (Chapter 2.1, bottom of p. 79) G/O [17] (Chapter 2, Exercise 2.1.1.14, p. 48) Gor [21] (Solution to Exercise 5.4, p. 296) Hob [27] (Sections 227 & 239, pp. 304 & 318) K/K [29] (Chapter 1, Theorem 1.1.3, pp. 2021) Kha [31] (Chapter 2, Theorem 1, p. 56 + middle p. 57) Nag [39] (Chapter 2, Section 2.4.1, pp. 9091) Oxt [40] (Theorem 7.8, p. 35) Ran [43] (Chapter 6, Section 8, Theorem 1, pp. 337338) R/S [44] (Theorem 1.2, p. 12) Spr [47] (Section 26, Corollary 26.6, p. 175) Tor [49] (Chapter 3.4, Exercise 4.4, p. 41; solution on p. 373)
PROOFS OF 1':
Abb [1] (Chapter 4.7, outlined on p. 128) Bea [2] (Chapter 8, Example 8.1.4, pp. 118119) Boa [4] (Section 22, pp. 159160) BBT [7] (Exercise 1:3.15 gives a construction to be verified) BBT [8] (end of Section 5.9.2, p. 248) B/K [10] (Chapter 131, Example 131, pp. 275276) Car [11] (end of Chapter 2 material and Exercise 34, pp. 3233) Cha [12] (Chapter 5, Section 3, Exercise C, p. 206) G/O [16] (Chapter 2, Example 18, p. 28) G/O [17] (Chapter 2, Exercise 2.1.1.14, pp. 4849) Hau [24] (Chapter 9, Section 42.3, p. 285; for rationals) Jef [27] (Chapter 5, Exercise 5.7, p. 138) K/K [29] (Chapter 1, Exercises 2 & 8, pp. 34 & 35) Kha [31] (Chapter 2, Exercise 2, pp. 5758) Kul [34] (Chapter 4, Section 8, Exercise 4, p. 133) Nag [39] (Chapter 2, Section 2.4.2, Theorem, pp. 9293) Oxb [40] (Theorem 7.8, p. 35) Pie [41] (Section 467, pp. 462463) Ran [43] (Chapter 6, Section 8, Problem 1, p. 338) R/S [44] (Theorem 1.2 & Corollary 1.3, p. 12) Spr [47] (Section 26, Example 26.7, pp. 175176) Tor [49] (Chapter 3.4, Exercise 4.4, p. 41; solution on p. 373)
HISTORY: I believe Ludwig Scheeffer (1884) was the first to obtain Theorem 1'. See pp. 7475 of Hawkins' "Lebesgue's Theory of Integration".
For a detailed discussion of Dini derivates of monotone functions, see the following post.
ESSAY ON NONDIFFERENTIABILITY POINTS OF MONOTONE FUNCTIONS http://groups.google.com/group/sci.math/msg/1bd39d992c91e950
PROOFS OF 2:
Abb [1] (Chapter 4.6, outlined on pp. 126127) Ben [3] (Chapter 1.3.1, Proposition 1.17, p. 24) BBT [8] (Chapter 6.7.2, Theorem 6.28, p. 277) Car [11] (Chapter 9, Theorem 9.2, p. 129) C/S [14] (Theorem 3.3) For [15] (Chapter 2.1, Theorem 2.6, pp. 8284) Gof [18] (Chapter 7, Section 6, Theorem 5', p. 89) Gol [19] (Chapter 5.6, Theorem 5.6E, p. 144) G/L/M/P [20] (Chapter 3, Problem 1.7, p. 26; solution on p. 168) Hau [24] (Chapter 9, Section 42.3, pp. 284285) K/N [28] (Problem 1.7.14, p. 33; Solution on p. 202) K/S [36] (Section 3.1, pp. 456457) Mit [38] (Exercise 3.11a, pp. 2425) Oxb [40] (Theorem 7.1, p. 31) Ran [43] (Chapter 6, Section 8, Problem 2, p. 338) R/S [44] (Section 7, Theorem 7.5, p. 44) Sie [45] (Chapter 7, Section 4, Example 8, p. 214) Sie [46] (Chapter 6, Section 72, pp. 183185) Sri [48] (Chapter 2, Section 2, 2.2.3 & 2.2.4, p. 54)
PROOFS OF 2':
BBT [8] (Section 6.7.2, Theorem 6.28, pp. 277279) For [15] (Chapter 2.1, Theorem 2.6, pp. 8284) G/O [16] (Chapter 2, Example 23, pp. 3031) G/O [17] (Chapter 2, Exercise 2.1.1.12, pp. 4849) G/L/M/P [20] (Chapter 3, Problem 1.8b, p. 26; solution on p. 168) Hah [22] (Chapter 3, Section 3, Satz V, pp. 201202) Hah [23] (Chapter 3, Section 26, Theorem 26.4.3, pp. 193194) Hob [25] (Section 237, pp. 314316) K/N [28] (Problem 1.7.16, p. 34; Solution on p. 203) Mit [38] (Exercise 3.11b, pp. 2425) Oxb [40] (Theorem 7.2, pp. 3132) Pie [41] (Section 473, pp. 467468) R/S [44] (Section 7, Exercises 7.G & 7.H, p. 45 gives outline) Sie [46] (Chapter 6, Section 72, Example, p. 185)
Kim [32] (see also C/S [14]) proves Theorem 2' for metric spaces having no isolated points (more generally, for F_sigma sets not containing any points isolated in the metric space) and functions f:X > R. [They were apparently unaware that the result in Hahn's 1932 book [23] (pp. 193194) is also for metric spaces.] Bolstein [5] proves a generalization of Hahn and Kim's result for a class of topological spaces that includes first countable spaces, locally compact Hausdorff spaces, separable spaces, and topological linear spaces. Chen/Su [13] prove that if X is a topological space, then every F_sigma subset of X is a discontinuity set for some function f:X > R if and only if there exists an everywhere discontinuous realvalued function on X. Mitsis [38] proves Theorem 2' for any function f:X > Y such that X is a topological space, Y is a metric space, and there exists a dense subset of X that has a dense complement.
HISTORY: Theorem 2' was first proved by William H. Young in 1903 [51] for functions f:R > R [and apparently independently by Lebesgue [37] (pp. 235236) in 1904], and generalized in 1905 [52] (pp. 376377) to functions f:R^n > R. The proof that Young gave in 1903 (in German) is very similar to the exposition (in English) that can be found on pp. 314316 of Hobson [25].
PROOFS OF 3:
Boa [4] (Section 18, pp. 123126) BBT [7] (Chapter 1.6, Theorem 1.19, pp. 2223) BBT [8] (Section 9.8, Theorem 9.39, pp. 422423) Car [11] (Chapter 11, Theorem 11.20, p. 183) For [15] (Chapter 4.1, Theorem 4.7, pp. 163164) Gor [21] (Theorem 5.16, pp. 7778; Solution to Exercise 5.9, p. 298) Hau [24] (Chapter 9, Section 42.4, pp. 286287) Hob [25] (Section 231, pp. 309310; for semicontinuous functions) Hob [26] (Sections 185 & 190, pp. 264265 & 273274) K/N [28] (Problem 1.7.20, p. 34; Solution on pp. 205206) Kec [30] (Section 24, Theorem 24.14, p. 193) Kur [35] (Chapter 2, Section 31, Theorem 1, p. 394; pp. 397398) K/S [36] (Section 4.3, pp. 460462) Mit [38] (Exercise 3.13a, p. 25) Oxt [40] (Theorem 7.3, p. 32) Pug [42] (Chapter 3, Section 3, Exercises 18 & 22 & 23, pp. 189190) R/S [44] (Section 11, Theorem 11.4, pp. 6768) Tow [50] (Chapter 3, Section 28, Theorem 4, pp. 130131)
REMARK: In the presence of Theorem 2, it suffices to show that C(f) is dense in R.
HISTORY: This result was obtained independently by William F. Osgood (1897) and René Baire (1899). http://mathforum.org/kb/thread.jspa?messageID=243385
PROOFS OF 3':
(semicontinuous result) Gof [18] (Chapter 7, Exercise 7.5, p. 98 states the result) Gor [21] (Solution to Exercise 5.18, pp. 302303) http://groups.google.com/group/sci.math/msg/72060bf0e6c2dae9
(bounded derivative result) Ben [3] (Chapter 1.3.2, Proposition, 1.10, p. 30) Bru [6] (Chapter 3, Section 2, Theorem 2.1, p. 34) B/L [9] (Theorem at bottom of p. 27) Gof [18] (Chapter 9, Exercise 2.3, p. 120 states the result) K/W [33]
HISTORY: Regarding the bounded derivative result, Bruckner and Leonard [9] (bottom of p. 27) wrote the following in 1966: "Although we imagine that this theorem is known, we have been unable to find a reference." I have found the result given in Exercise 2.3 on p. 120 of a 1953 text by Casper Goffman, but nowhere else prior to 1966 (including Goffman's Ph.D. Dissertation).
PROOFS OF 4 & 4':
References omitted because this follows from Theorem 2 and the fact  whose proof can be found in most any real analysis text  that a bounded function f:[a,b] > R is Riemann integrable if and only if D(f) has measure zero.
Regarding our more precise (and apparently stronger for one direction) version, note that any F_sigma measure zero set is a countable union of closed measure zero sets, and hence a countable union of nowhere dense sets.
Interestingly, despite the ease in which this more precise version follows from results in virtually every graduate level real analysis text, I have not seen this more precise version explicitly stated outside of a handful of research papers. Rooij/Schikhof [44] comes the closest that I've seen. Their Exercise 6.I (p. 42) asks the reader to verify that every F_sigma measure zero set is meager and their Theorem 7.5 (p. 44) implies that D(f) is F_sigma. Even Oxtoby [40], which gives an extensive overview of various measure and category analogs, doesn't mention that D(f) is meager for a Riemann integrable f, despite having (1) a proof of the Riemann integrability continuity condition (pp. 3334), (2) a proof that any D(f) set is F_sigma (p. 31), and (3) the observation that any F_sigma Lebesgue measure set is meager (bottom of p. 51).
What makes this result more interesting is that for a Riemann integrable function f, D(f) is actually "infinitely smaller than" some meagerandmeasurezero sets. More precisely, there exists a set E such that E is meager and E has measure zero such that E cannot be covered by countably many F_sigma measure zero sets (the latter being the discontinuity sets of Riemann integrable functions). Thus, not only is it an understatement to describe the size of the discontinuity sets of Riemann integrable functions by saying they have measure zero (because they are also small in the Baire category sense), but it's even an understatement to describe their size by saying they are simultaneously measure zero and meager! For more about the size classification that discontinuity sets of Riemann integrable functions belong to, see the following post.
HISTORICAL ESSAY ON F_SIGMA LEBESGUE NULL SETS http://groups.google.com/group/sci.math/msg/00473c4fb594d3d7

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Dave L. Renfro

