ESSAY ON NON-DIFFERENTIABILITY POINTS OF MONOTONE FUNCTIONS
Dave L. Renfro
[sci.math Wed, 01 Nov 2000 08:13:50 +0100]
<http://forum.swarthmore.edu/epigone/sci.math/grinqualven>
wrote (in part):
> Therefore f is not differentiable on a dense set of I.
> If I'm not mistaken, since f is monotone, continuous it can be
> non-differentiable only on a Lebesgue-Zero-set.
>
> My question now is: What is the largest subset of I,
> where f is not differentiable? What else can be said about f.
> It is easy to construct and plot, B.T.W.
In what follows, upper case letters in parentheses refer to
theorems and lower case letters in parenthesis refer to examples.
[I mean the colloquial use of "theorem" and "example", in case
sci.math's watchdog is reading this.] All functions are assumed
to be defined on [0,1] and the terms "complement", "residual",
etc. are relative to [0,1] unless otherwise noted.
Theorem (A) was proved by Lebesgue in 1904 (his Ph.D. thesis
and the first edition of his book "Lecons Sur L'Integration")
for continuous monotone functions. In 1911 Young [13] proved that
(A) (with a different proof than Lebesgue used, I believe)
holds regardless of any continuity conditions. [Of course, any
monotone function is continuous except for a countable set.
(Any countable set can be the set of discontinuities of a monotone
function, even of a strictly increasing function, though.)]
(A) For any monotone function f, continuous or not,
{x: f'(x) does not exist OR f'(x) is infinite} has Legesgue
measure zero.
Theorem (A) is sharp, even in the presence of continuity:
(a) For any set E having Lebesgue measure zero there exists a
strictly increasing continuous function f on [0,1] such
that f'(x) = infinity for all x in E. In particular, if E is
chosen to be residual in [0,1] (i.e. the relative complement
of E in [0,1] is a first category set), then f'(x) = infinity
except for a first category set of x's in [0,1]. [Note that
the "strictly increasing" part is easy to get if you can find
a nondecreasing example, since we can use f(x) = g(x) + x if
g(x) is a nondecreasing example.]
Example (a) can be found in [2, pp. 288-289], [5, p. 62],
[6, pp. 214-215], and [9, 25-26].
Let d-, d+, D-, and D+ denote the lower left, lower right,
upper left, and upper right Dini derivates of f: [0,1] --> reals.
Recall that each Dini derivate function is defined on (0,1) (and
two of the four are defined at each endpoint) and they take values
in {reals} union {-infinity, +infinity} (i.e. the extended real
line). For an arbitrary function we have d- less= D- and
d+ less= D+. Also, f' exists and is finite at x=a <==>
-infinity < d- = D- = d+ = D+ < +infinity at x=a.
Young [11] proved the following in 1908:
(B) For an arbitrary function the set {x: (D- < d+) or (D+ < d-)}
is countable.
Thus, if we want to get to get strictly increasing continuous
functions having large sets of non-differentiability, we need
to look at other ways for the Dini derivates not to have the
same finite value. One of these ways has generated several
interesting results. Specifically, I'll be considering the set
{x: (d- not= d+) or (D- not= D+)}, which is in general a proper
subset of {x: f'(x) does not exist OR f'(x) is infinite}. The next
two examples show that even for strictly increasing continuous
functions the set {x: (d- not= d+) or (D- not= D+)} can be much
smaller than the set {x: f'(x) does not exist OR f'(x) is infinite}.
Indeed, the former set can be first category when the latter set
has a first category complement.
(b) There exists a strictly increasing continuous function f on
[0,1] such that d- = d+ = 0 and D- = D+ = 1 for all except
a first category set of x's in [0,1]. Given 0 < a < b, let
g(x) = (b-a)*f(x) + ax. Then g is a strictly increasing
continuous function satisfying d- = d+ = a and D- = D+ = b
for all except a first category set of x's in [0,1]. The
first example can be found in Evans/Humke [4, p. 257] and
both examples can be found in Zajicek [14, pp. 204-205].
Zajicek speculates that it is due to Z. Zahorski (probably
mid to late 1940's).]
(c) Let a > 0 and h(x) = (f^{-1})(x) + (a-1)*x, where f^{-1} is
the inverse of the function f(x) in example (b). Then h
is a strictly increasing continuous function satisfying
d- = d+ = a and D- = D+ = infinity for all except a first
category set of x's in [0,1]. This is example 2 in Evans/Humke
[4, p. 205] and is (essentially) example 3 in Zajicek
[14, p. 258].
In both examples (b) and (c) the set {x: (d- not= d+) or (D- not= D+)}
is a first category set. It turns out that for continuous functions
we have no choice about this. Young [12, p. 305] gave *two* proofs
of theorem (C) in 1908.
(C) For any continuous function, monotone or not, the set
{x: (d- not= d+) or (D- not= D+)} is first category.
Theorem (C) was rediscovered by Neugebauer [7] in 1962 (with a
proof different from either of those that Young gave), who was
unaware of Young's earlier paper.
By combining theorems (A) and (C), we see that the set
{x: (d- not= d+) or (D- not= D+)} is small in both the measure
sense and in the topological sense:
(D) For any continuous monotone function, the set
{x: (d- not= d+) or (D- not= D+)} is simultaneously first
category and measure zero.
Theorem (D), even when "continuous" is omitted, was strengthened
independently by Evans/Humke [4] and Zajicek [14] (it's an
immediate corollary of Zajicek's theorem 1):
(E) For any monotone function, the set
{x: (d- not= d+) or (D- not= D+)} is sigma-porous.
I'll probably post some stuff about porous and sigma-porous
sets one day, but for now I'll just mention these facts:
Every sigma-porous set is simultaneously first category and
measure zero. There exists a set simultaneously first category and
measure zero that is not contained in any sigma-porous set
(equivalently, that is not contained in any countable union of
sigma-porous sets). Indeed, there even exists a closed measure
zero set having this property.
Thus far, nothing has been said about the fractal dimensions
of these exceptional sets. Vallin [10] showed in 1991 that the
set {x: D- not= D+} can have Hausdorff dimension 1 for a strictly
increasing continuous function (in particular, a strictly increasing
continuous function can be non-differentiable on a set having
Hausdorff dimension 1), EVEN when {x: D- not= D+} is very small
in the sense of porosity. [More precisely, the set in question
having Hausdorff dimension 1 is strongly symmetrically porous.
The significance in my usage of "EVEN" is due to the fact that
there exist closed porous sets that are not contained in any
countable union of strongly symmetrically porous sets.]
(d) There exists a strictly increasing continuous function such
that {x: D- not= D+} has Hausdorff dimension 1.
Incidentally, while this implies that the set
{x: (d- not= d+) or (D- not= D+)} has Hausdorff dimension 1,
there's no genuine improvement in getting the smaller set
{x: D- not= D+} to have Hausdorff dimension 1. This is because
if you have a function such that {x: (d- not= d+) or (D- not= D+)}
has Hausdorff dimension 1, then at least one of the sets
{x: D- not= D+} and {x: d- not= d+} must have Hausdorff
dimension 1. [And if it's not {x: D- not= D+}, use -1 times
your function.]
Note that this is the strongest possible fractal dimension result,
since "Hausdorff dimension 1" implies that all the other dimensions
(packing, upper box, lower box, etc.) are 1 as well. Although
example (a) prevents us from being able to establish any
"logarithmically less than 1" Hausdorff dimension result for
{x: f'(x) does not exist OR f'(x) is infinite}, I don't know if this
is the case for the smaller set {x: (d- not= d+) or (D- not= D+)}.
Because any bounded variation (BV) function is the difference of
two monotone functions, theorem (A) holds for BV functions. This
is because differentiation distributes over pointwise subtraction,
and so the non-differentiability points for f-g are a subset of
(non-differentiability points of f) UNION (non-differentiability
points of g). On the other hand, the property of not belonging to
the set {x: (d- not= d+) or (D- not= D+)} does not distribute over
pointwise subtraction, and so the analogous set for f-g might not
be contained in the union of the corresponding sets for f and g.
I believe this is what allows for examples (e) and (f). Note that
while theorem (E) says that the set {x: (d- not= d+) or (D- not= D+)}
is small for monotone functions, even in the absence of continuity,
the next example shows that it can be much larger for BV functions,
even in the presence of continuity.
(e) Let E be any subset of an F_sigma measure zero set. Then
there exists a continuous BV function f such that
|D+ - D-| > 1 for each x in E.
Example (e) appears as theorem 3 in Evans/Humke [4]. [They
state that E can be any first category and measure zero set,
but their proof only establishes the result for subsets of
F_sigma measure zero sets.] Any subset of an F_sigma measure
zero set is automatically first category and measure zero, but
the converse is not true. More discussion of this matter, and
a proof of this fact, can be found in my sci.math post at
HISTORICAL ESSAY ON F_SIGMA LEBESGUE NULL SETS [May 1, 2000]
<http://forum.swarthmore.edu/epigone/sci.math/dwillydwul>.
Recall that I said there exists a closed measure zero set
that is not sigma-porous (the discussion following theorem (E)).
This means that some continuous BV functions can have
{x: (d- not= d+) or (D- not= D+)} larger (in one sense) than is
possible for any monotone function. [Technical note: Unfortunately,
there also exist sigma-porous sets--even porous sets--that are not
subsets of any F_sigma measure zero set. So neither of the
"yardsticks" in theorem (E) and example (e) is stronger than the
other. I suspect that the exceptional set in theorem (E) is
actually a subset of an F_sigma sigma-porous set, which would
then make it strictly smaller than the exceptional set that shows
up in example (e), but I haven't investigated this yet.]
The property of being absolutely continuous is stronger
than "continuous BV". Thus, example (f) is an improvement over
example (e).
(f) Let E be any subset of an F_sigma measure zero set. Then
there exists an absolutely continuous function f
such that |D+ - D-| > 1 for each x in E.
The reference for this is the same as that for example (e), since
the continuous BV function Evans/Humke [4] construct is actually
absolutely continuous. (This is pointed out on p. 259 of [1].)
Since I'm a bit partial to Baire category results involving
pathology, I'll end with a few on this topic that I'm aware of.
Zamfirescu [15] [16] proved that MOST nondecreasing continuous
functions on [0,1] are not differentiable at MOST points, when
"MOST" is used the Baire category sense in both places. More
precisely, he proved the following.
(F) Let CND[0,1] be the collection of nondecreasing continuous
functions with the sup norm. Then for all f in CND[0,1] except
a first category set of f's we have:
(F1) (d- = d+ = 0) and (D- = D+ = infinity) for all except a first
category set of x's in [0,1]. [16]
(F2) (d- = 0 or D- = infinity) for each x in (0,1]. [15]
(F3) (d+ = 0 or D+ = infinity) for each x in [0,1). [15]
(F4) f'(x) = 0 for all except a measure zero set of x's. [15]
Note that (F2) and (F3) imply that at no point is there a positive
finite unilateral derivative. [Cater [3] recently gave a nice
construction of a function satisfying (F2) and (F3).]
Petruska [8] recently announced the following interesting theorems.
(G) Let E be an analytic subset of [0,1]. Then the set of
nondecreasing functions (not necessarily continuous) which
are not finitely differentiable at any point of E is residual
in the space of nondecreasing functions (sup norm) if and only
if E is contained in an F_sigma Lebesgue measure zero set.
(H) Let E be an analytic subset of [0,1]. Then the set of
absolutely continuous functions which are not finitely
differentiable at any point of E is residual in the space of
absolutely continuous functions (variational norm) if and only
if E is contained in an F_sigma Lebesgue measure zero set.
Note that the set E is to be specified before obtaining the
residual sets in (G) and (H). However, by intersecting countably
many residual sets (any such intersection will still be residual),
we can replace E with any countable union of sets, each of which
is contained in some F_sigma Lebesgue measure zero set. Nonetheless,
this gives no improvement in either (G) or (H), since it is easy
to see that the property of being a subset of some F_sigma Lebesgue
measure zero set is preserved under countable unions.
[1] Charles L. Belna, Gerald T. Cargo, Michael J. Evans, and Paul
D. Humke, "Analogues of the Denjoy-Young-Saks Theorem", Trans.
Amer. Math. Soc. 271 (1982), 253-260.
[2] Andrew M. Bruckner, Judith B. Bruckner, and Brian S. Thomson,
REAL ANALYSIS, Prentice-Hall, 1997.
[3] F. S. Cater, "On the Dini derivates of a particular function",
Real Analysis Exchange 25 (1999-2000), 943-946.
[4] Michael J. Evans and Paul D. Humke, "The equality of unilateral
derivates", Proc. Amer. Math. Soc. 79 (1980), 609-613.
[5] A. B. Kharazishvili, STRANGE FUNCTIONS IN REAL ANALYSIS,
Marcel Dekker, 2000.
[6] I. P. Natanson, THEORY OF FUNCTIONS OF A REAL VARIABLE,
Volume I, Frederick Ungar Publishing Company, 1955.
[7] Christoph J. Neugebauer, "A theorem on derivates", Acta Sci.
Math. (Szeged) 23 (1962), 79-81.
[8] Gyorgy Petruska, "Points of non-differentiability of
differentiable functions", Real Analysis Exchange 24
(1998-99), 79-80. [Abstract of a talk given in June 1998.]
[9] A. C. M. Van Rooij and W. H. Schikhof, A SECOND COURSE ON
REAL FUNCTIONS, Cambridge Univ. Press, 1982.
[10] Robert W. Vallin, "Symmetric porosity, dimension, and
derivates", Real Analysis Exchange 17 (1991-92), 314-321.
[11] William H. Young, "On the distinction of right and left at
points of discontinuity", Quarterly Journal of Pure and
Applied Mathematics 39 (1908), 67-83.
[12] William H. Young, "Oscillating successions of continuous
functions", Proc. London Math. Soc. (2) 6 (1908), 298-320.
[13] William H. Young, "On functions of bounded variation",
Quarterly Journal of Pure and Applied Mathematics 42 (1911),
54-85.
[14] Ludek Zajicek, "on the symmetry of Dini derivates of arbitrary
functions", Comment. Math. Univ. Carolinae 22 (1981), 195-209.
[15] Tudor Zamfirescu, "Most monotone functions are singular",
Amer. Math. Monthly 88 (1981), 47-49.
[16] Tudor Zamfirescu, "Typical monotone continuous functions",
Arch. Math. 42 (1984), 151-156.
Dave L. Renfro