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Topic: References for Continuity Sets
Replies: 2   Last Post: Dec 20, 2006 4:23 PM

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Dave L. Renfro

Posts: 4,792
Registered: 12/3/04
Re: References for Continuity Sets
Posted: Dec 20, 2006 4:23 PM
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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 real-valued 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, semi-continuous 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, c-dense in R, and even co-meager
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
semi-continuous or to be a bounded derivative.

REMARK: A meager F_sigma set can be c-dense in R, have a
measure zero complement, or to even have a Haudsorff
h-measure zero complement for any pre-assigned 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 13-1, 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. 20-21)
Kha [31] (Chapter 2, Theorem 1, p. 56 + middle p. 57)
Nag [39] (Chapter 2, Section 2.4.1, pp. 90-91)
Oxt [40] (Theorem 7.8, p. 35)
Ran [43] (Chapter 6, Section 8, Theorem 1, pp. 337-338)
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. 118-119)
Boa [4] (Section 22, pp. 159-160)
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 13-1, Example 13-1, pp. 275-276)
Car [11] (end of Chapter 2 material and Exercise 34, pp. 32-33)
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. 48-49)
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. 57-58)
Kul [34] (Chapter 4, Section 8, Exercise 4, p. 133)
Nag [39] (Chapter 2, Section 2.4.2, Theorem, pp. 92-93)
Oxb [40] (Theorem 7.8, p. 35)
Pie [41] (Section 467, pp. 462-463)
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. 175-176)
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. 74-75 of Hawkins'
"Lebesgue's Theory of Integration".

For a detailed discussion of Dini derivates of monotone
functions, see the following post.

ESSAY ON NON-DIFFERENTIABILITY POINTS OF MONOTONE FUNCTIONS
http://groups.google.com/group/sci.math/msg/1bd39d992c91e950


PROOFS OF 2:

Abb [1] (Chapter 4.6, outlined on pp. 126-127)
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. 82-84)
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. 284-285)
K/N [28] (Problem 1.7.14, p. 33; Solution on p. 202)
K/S [36] (Section 3.1, pp. 456-457)
Mit [38] (Exercise 3.11a, pp. 24-25)
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. 183-185)
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. 277-279)
For [15] (Chapter 2.1, Theorem 2.6, pp. 82-84)
G/O [16] (Chapter 2, Example 23, pp. 30-31)
G/O [17] (Chapter 2, Exercise 2.1.1.12, pp. 48-49)
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. 201-202)
Hah [23] (Chapter 3, Section 26, Theorem 26.4.3, pp. 193-194)
Hob [25] (Section 237, pp. 314-316)
K/N [28] (Problem 1.7.16, p. 34; Solution on p. 203)
Mit [38] (Exercise 3.11b, pp. 24-25)
Oxb [40] (Theorem 7.2, pp. 31-32)
Pie [41] (Section 473, pp. 467-468)
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. 193-194) 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 real-valued 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. 235-236) in 1904],
and generalized in 1905 [52] (pp. 376-377) 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. 314-316 of
Hobson [25].


PROOFS OF 3:

Boa [4] (Section 18, pp. 123-126)
BBT [7] (Chapter 1.6, Theorem 1.19, pp. 22-23)
BBT [8] (Section 9.8, Theorem 9.39, pp. 422-423)
Car [11] (Chapter 11, Theorem 11.20, p. 183)
For [15] (Chapter 4.1, Theorem 4.7, pp. 163-164)
Gor [21] (Theorem 5.16, pp. 77-78; Solution to Exercise 5.9, p. 298)
Hau [24] (Chapter 9, Section 42.4, pp. 286-287)
Hob [25] (Section 231, pp. 309-310; for semi-continuous functions)
Hob [26] (Sections 185 & 190, pp. 264-265 & 273-274)
K/N [28] (Problem 1.7.20, p. 34; Solution on pp. 205-206)
Kec [30] (Section 24, Theorem 24.14, p. 193)
Kur [35] (Chapter 2, Section 31, Theorem 1, p. 394; pp. 397-398)
K/S [36] (Section 4.3, pp. 460-462)
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. 189-190)
R/S [44] (Section 11, Theorem 11.4, pp. 67-68)
Tow [50] (Chapter 3, Section 28, Theorem 4, pp. 130-131)

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':

(semi-continuous result)
Gof [18] (Chapter 7, Exercise 7.5, p. 98 states the result)
Gor [21] (Solution to Exercise 5.18, pp. 302-303)
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. 33-34), (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 meager-and-measure-zero 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




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