Differentiability of the Ruler Function
Dave L. Renfro
irrational, f(0) = 1, and f(x) = 1/q if x = p/q
where p and q are relatively prime integers with q > 0.
It is well-known that f is continuous at each irrational
point and discontinuous at each rational point.
On November 9, 2005 I posted some references about the
differentiability of ruler-like functions.
http://groups.google.com/group/sci.math/msg/cee496c760dd15be
That post, although rather detailed for the topic,
was hurriedly made. The present post is an extensive
revision of that post.
The ruler function itself seems to have been first
introduced on pp. 14-15 of an 1875 book by Thomae:
Carl Johannes Thomae, EINLEITUNG IN DIE THEORIE DER
BESTIMMTEN INTEGRALE, Halle/Saale: Nebert, 1875.
[JFM 7.0153.01]
http://www.mathematik.uni-halle.de/history/thomae/index.html
http://tinyurl.com/yxl78j [google translated version]
I have not been able to find a digitized copy of this
book on the internet, nor have I come across a copy
in a library or elsewhere, but I have seen a few references
to this in the literature (none of the items below
mention the origin of the ruler function). For example,
the Thomae reference is mentioned on p. 52 of Thomas
Hawkins' book "Lebesgue's Theory of Integration".
Here is a summary of the main results below.
In this summary, f always refers to the ruler
function as defined above.
** f is nowhere differentiable.
We would expect higher powers of f to be smoother,
and this is what we find. Note that for each r > 0,
the sets where f^r is continuous and discontinuous
is the same as for f.
** For each 0 < r <= 2, f^r is nowhere differentiable.
** For each r > 2, f^r is differentiable on a set that
has c many points in every interval.
The results above can be further refined.
** For each 0 < r < 2, f^r satisfies no pointwise
Lipschitz condition. Heuer [15]
** For r = 2, f^r is nowhere differentiable and
satisfies a pointwise Lipschitz condition on
a set that is dense in the reals. Heuer [15]
** For r > 2, f^r is differentiable on a set whose
intersection with every open interval has Hausdorff
dimension 1 - 2/r. Frantz [20]
Using ruler-like functions that "damp-out" quicker
than any power of f gives behavior that one would
expect from the above.
Let w:Z+ --> Z+ be an increasing function that
eventually majorizes every power function. Define
f_w(x) = 0 for x irrational, f_w(0) = 1, and
f_w(p/q) = 1/w(q) where p and q are relatively
prime integers.
** f_w is differentiable on a set whose complement
has Hausdorff dimension zero. Jurek [4] (pp. 24-25)
Interesting, each of the sets of points where these
functions fail to be differentiable is large in the
sense of Baire category.
THEOREM: Let g be continuous and discontinuous on sets
of points that are each dense in the reals.
Then g fails to have a derivative on a
co-meager (residual) set of points. In fact,
g fails to satisfy a pointwise Lipschitz
condition, a pointwise Holder condition,
or even any specified pointwise modulus of
continuity condition on a co-meager set.
(Each co-meager set has c points in every interval.)
There are 22 items below. I found 4 of them on the internet,
I provide the complete text for 9 of them, and I give
some idea of what the remaining 9 items involve.
On the internet -- [2], [4], [11], [22].
Text provided below -- [1], [3], [5], [6], [12], [13],
[14], [19], [21].
---------------------------------------------------------------
[1] Wilhelmus David Allen Westfall, "A class of functions
having a peculiar discontinuity", Bulletin of the American
Mathematical Society 15 (1908-09), 225-226.
[JFM 40.0437.02] [Submission date: 12 December 1908]
The complete text of the paper follows, with minor
editing changes to accommodate ASCII format.
Consider all functions discontinuous for all rational values
of the independent variable, and continuous and equal to zero
for all irrational values. They are of the form
f(p/q) /= 0, p and q prime to each other,
(1) f(b) = 0, for b irrational, with the condition that
Lim(q --> oo) f(p/q) = 0.
The following are examples of such functions:
(2) g_n(p/q) = 1/q^n, g_n(1) = 1, g_n(b) = 0,
(3) g(p/q) = 1/q!, g(1) = 1, g(b) = 0, [*]
(4) h(p/q) = 1/q^q, h(1) = 1, h(b) = 0.
[*] An example first used by Professor Osgood in his lectures.
Liouville has shown [Comptes Rendus for 1844] that if b is
an algebraic irrationality of the n'th order
| p/q - b | > A/q^n,
where A is independent of q. Hence g_n has a zero derivative
at all algebraic irrationalities of order less than n, and
g and h at all algebraic irrationalities.
It will now be shown that it is impossible for a function
of type (1) to have a derivative for every irrational value
of the independent variable. Let b_1, b_2, b_3, ... be an
infinite sequence of rational numbers such that
| b_n - b_(n-1) | < 1 / |f(b_n)| and < |1/q^q|,
if b_n = p/q. Then this defines a transcendental number b
at which the difference quotient of f(x) taken over the
sequence b_n is greater than 1/2. Hence a derivative cannot
exist, since it is evidently zero if it exists. The definition
of this point shows that such a point exists in every interval.
The above holds equally well for a function discontinuous at
all rational points, and continuous at all irrational points
in such a way that it coincides in these points with a function
having a derivative throughout.
---------------------------------------------------------------
[2] Franz Lukács, "Eine unstetige und differenzierbare
Fundtion", Mathematische Annalen 70 (1911), 561-562.
[JFM 42.0420.02] [Submission date: 1 November 1910]
Appears to prove the same thing that Westfall did
for the specific example of f(p/q) = (1/q)^q.
This paper is on the internet at
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D36758.html
---------------------------------------------------------------
[3] Tsuruichi Hayashi, "Eine stetige und nicht-differenzierbare
function", Tôhoku Mathematical Journal 1 (1911-12),
140-142. [JFM 43.0482.03] [No submission date given.]
Hayashi mentions Lukács' paper. I'm not sure if Hayashi
is filling in some gaps from Lukács' paper or extending
the results in Lukács' paper in some way. Hayashi's paper
is in German, which I can't read. [Lukács' paper is also
in German, but in that case it was easy to figure out what
Lukács was doing. In this case, since Hayashi already knows
of Lukács' paper, the issue of what Hayashi is doing is
not as immediately apparent to me.]
The complete text of the paper follows, with minor
editing changes to accommodate ASCII format.
Im 70. Bande der Mathematische Annalen, S. 561, 1911, finden
wir ein einfaches von Herrn Franz Lukács gegebenes Beispiel
einer Funktion, die in einer überall dichten Menge unstetig
und doch in einer anderen überall dichten Menge differenzierbar
ist. Nach der Lukács-schen Methode, gebe ich im folgenden
ein sehr einfaches Beispiel einer Funktion, die in einer
überall dichten Menge stetig uud nichtdifferenzierbar ist.
Mein Beispiel wird als ein Resultat des Satzes von Liouville
deduziert, wie Herr Lukács's Beispiel.
Wer definieren die Funktion f(x) wie folgt: Für jedes
irrationale x sei f(x) = 0; wenn x rational und auf den
kleinsten positiven Nenner gebracht = p/q ist, so sei
f(x) = f(p/q) = 1/q.
Dann ist wie leicht ersichtlich, die so definierte
Funktion für jeden rationalen Wert von x and also
in einer überall dichten Menge unstetig, und doch
für jeden irrationalen Wert von x stetig. Die
Funktion f(x) ist für jeden nicht-algebraischen,
i.e. transzendentalen Wert von x, der ein Element
der von Liouville angegebenen Menge ist, und also
in einer überall dichten Menge, (1) nicht-differenzierbar.
(1) Vgl. A. Schönflies: Die Entwickelung der Lehre von
Punktmannigfaltigkeiten, Jahresbaricht der Deutschen
Mathematiker-Vereinigung. 8ter Band, S. 103, 1900.
Für den Beweis bilden wir den Differenzenquotient
H = (x,b) = [f(x) - f(b)]/(x-b) = f(x)/(x-b),
wo b ein Element Liouvillescher Menge ist.
Wenn x irrational ist, so haben wir H(x,b) = 0.
Wenn x rational und auf den kleinsten positiven Nenner
gebracht = p/q ist, so haben wir
H(x,b) = H(p/q, b) = (1/q) / (p/q - b).
Nun für die transzendentale Zahl b, ist
| p/q - b | < 1/(Mq^n),
wo n >= 2 ist.
Wenn p/q - b > 0, i.e. als den vorwärts genommenen
Differenzenquotient betrachtet, ist daher
H(p/q, b) > (1/q) / (1/Mq^n) = Mq^(n-1).
Der vorwärts genommene Differentialquotient ist also
positive und wird unendlich.
Wenn p/q - b < 0, i.e. als den rückwärts genommenen
Differenzenquotient betrachtet, ist
H(p/q, b) = (1/q) / (p/q - b) < 0
und ist
p/q - b > -1 / (Mq^n).
Also ist
H(p/q, b) < (1/q) / (-1/Mq^n) = -Mq^(n-1).
Daher ist der rückwärts genommene Differentialquotient
negativ und wird unendlich.
Die Funktion ist für alle Argumente nicht-differenzierbar,
nicht nur für transzendente Zahlen. Dar Beweis ist sehr
einfach folgender.
Sie b ein irrationaler Wert und x ein irrationaler
Nachbarwert, dann ist f(x) - f(b) = 0 und daher der
Differenzenquotient = 0. Andererseits lässt sich x
durch eine Reihe rationaler Zahlen, die Näherungsbrüche
des Kettenbruchs für b, in der weise annähern, dass,
wenn p_n/q_n ein solcher Näherungsbruch in reduzierter
Form ist, die Ungleichung besteht
| b - p_n/q_n | < 1 / (q_n)^2.
Daher wächst der Differenzenquotient
[ f(p_n/q_n) - f(b) ] / [ p_n/q_n - b ]
über alle Grenzen mit wachsendem n. Es kann daher kein
Differentialquotient existieren.
---------------------------------------------------------------
[4] Bohus Jurek, "Sur la dérivabilité des fonctions à
variation bornée", Casopis Pro Pestování Matematiky
a Fysiky 65 (1935), 8-27. [Zbl 13.00704; JFM 61.1115.01]
It appears that Jurek proves some general results
concerning the zero Hausdorff h-measure of
sets of non-differentiability for bounded
variation functions such that the sum of the
h-values of the countably many jump discontinuities
is finite (special case: h(t) = t^r for a fixed
0 < r < 1). General "h-versions" of the ruler
function seem to appear as examples, and V. Jarnik's
more precise results about the Hausdorff dimension
of Liouville-like Diophantine approximation results
are used.
This paper is on the internet at
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D98714.html
http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D98723
---------------------------------------------------------------
[5] Max Goy Scherberg, "On the graph of a certain function",
abstract of talk given 11 May 1940 at the annual meeting
of the Minnesota Section of the MAA, American Mathematical
Monthly 47 #7 (Aug./Sept. 1940), 436.
The complete text of the abstract follows, with minor
editing changes to accommodate ASCII format.
It is readily shown that the graph of the multi-valued
function f(x) = 0 for x irrational and in 0 <= x <= 1,
f(x) = 1/s for x = r/s a rational number in 0 <= x <= 1,
consists of the irrational point in 0 <= x <= 1 and the
intersections among the lines
y = x, y = x/2, y = x/3, ...
x = 0, y = 1, y = 1/2, y = 1/3, y = 1/4, ...
Dr. Scherberg then showed with a simple geometric argument
that f(x) is continuous at the irrational points.
---------------------------------------------------------------
[6] Jerry William Gaddum, "An interesting function: Solution
to Monthly Problem E 935", American Mathematical Monthly
58 #5 (May 1951), 340.
The complete text of the item follows, with minor
editing changes to accommodate ASCII format.
E 935 [1950, 557]. Proposed by C[arl] D[ouglas] Olds, San Jose
State College
Given [sic] an example of a function which is discontinuous in
an everywhere dense set, and which is also differentiable in an
everywhere dense set.
Solution by J. W. Gaddum, University of Missouri. We define
f(x) for x in [0,1]. If x = k/2^n, where k is an odd integer
less than 2^n, set f(x) = 1/2^(2n-1). Otherwise set f(x) = 0.
For 2^(n-1) values of x, f(x) = 1/2^(2n-1). Hence the variation
of f(x) on [0,1] is
SUM(n=1 to oo) [2^(n-1) / 2^(2n-1)] = SUM(n=1 to oo) 1/2^n = 1/2.
Hence f(x) is of bounded variation and is differentiable almost
everywhere. But f(x) is discontinuous for x = k/2^n.
Also solved by Leopold Flatto, Norman Miller, Leo Moser,
C. S. Ogilvy, L. A. Ringenberg, J. B. Rosser, O. E. Stanaitis,
Albert Wilansky, and the proposer.
Several solutions offered f(x) = 0 if x is irrational and
f(x) = q^(-q) if x = p/q, (p,q) = 1. Then f(x) is discontinuous
for every rational x and differentiable for every algebraic
irrationality. A very elementary, but perhaps somewhat
unsuportsmanlike, example is f(x) defined on [-1,1] by
f(x) = 1 for negative rational x and f(x) = 0 otherwise.
F. Bagemihl located this problem, with a solution, in
K. Knopp, Aufgabensammlung zur Funktionentheorie, vol. 2,
Berlin (1949), p. 9, prob. 2; p. 55.
---------------------------------------------------------------
[7] Simon Cetkovic, "Sur la différentiabilité des deux
familles des fonctions réelles", Bulletin de la Société
des Mathématiciens et Physiciens de la RP de Serbie
(Yougoslavie) 4 (1952), 53-57. [MR 14,855b]
MR 14,855b (E. Hewitt): "Let A and B be complementary
sets of natural numbers, each containing an infinite
number of primes. Let S be the set of real numbers
which can be written in the form p/q, where p is an
integer, q in B, and p and q are relatively prime.
Let F(x) = 0 on S' and let F(p/q) = q^(-alpha) for
p/q in S, alpha being an arbitrary positive real
number. Then, if 0 < alpha <= 1, F is nowhere
differentiable but is continuous on an everywhere
dense set; if 1 < alpha, then F is discontinuous
on a dense set but is also differentiable on a
dense set."
---------------------------------------------------------------
[8] Simon Cetkovic, "La relation entre l'ordre des
nombres algébriques et la différenciabilité d'une
famille des fonctions", Bulletin de la Société
des Mathématiciens et Physiciens de la RP de Serbie
(Yougoslavie) 5 (1953), 91-92. [MR 15,693d; Zbl 52.28803]
MR 15,693d (no reviewer cited): "Après avoir défini
une famille de fonctions de la manière suivante:
F(x,b) = 0, si x est un nombre irrationel,
F(x,b) = q^(-b), si x est un nombre rationnel
de la forme p/q, p et q étant des entiers sans
facteurs communs avec q /= 0 et où b est une
paramètre arbitraire, l'auteur démontre: La
fonction F(x,b) a une dérivée aux tous points
irrationnels et algébriques de l'ordre <= n
pour n < b, quoique elle ne soit pas continue
aux points d'un ensemble de points partout dense."
---------------------------------------------------------------
[9] Isaac Jacob Schoenberg, "The integrability of certain
functions and related summability methods", American
Mathematical Monthly 66 #5 (May 1959), 361-375.
[MR 21 #3696; Zbl 89.04002]
After observing that the usual ruler function is
Riemann integrable (it has countably many points
of discontinuity), Schoenberg proves the following
theorem: Let {b_n} be a sequence of real numbers
and define the ruler-like function f by putting
f(p/q) = b_q. Then f is Riemann integrable on
[0,1] if and only if b_n --> 0 as n --> oo.
Schoenberg does this in the first two pages.
The paper contains quite a bit more that is too
detailed and technical for me to concisely describe.
---------------------------------------------------------------
[10] Isaac Jacob Schoenberg, "The integrability of certain
functions and related summability methods II", American
Mathematical Monthly 66 #7 (Aug./Sept. 1959), 562-563.
[MR 21 #6411; Zbl 89.04002]
Some follow-up literature results that has come
to his attention, including the paper by Lukács
(but none of the other items listed above).
---------------------------------------------------------------
[11] Solomon Marcus, "Sur les propriétés différentielles
des fonctions dont les points de continuité
forment un ensemble frontiére partout dense",
Annales Scientifiques de l'École Normale Supérieure
(3) 79 (1962), 1-21. [MR 34 #2792; Zbl 105.04502]
A very thorough survey of the title's topic that
makes extensive use of Lebesgue measure and Baire
category to describe the size of the exceptional
sets involved. Many variations of the ruler function
are analyzed, and most of the references listed
above are cited (but not Jurek or Westfall).
This paper is on the internet at
http://www.numdam.org/numdam-bin/browse?id=ASENS_1962_3_79_1
---------------------------------------------------------------
[12] Gerald J. Porter, "On the differentiability of a
certain well-known function", American Mathematical
Monthly 69 (1962), 142. [MR1531547]
The complete text of the paper follows, with minor
editing changes to accommodate ASCII format.
The function given by f(x) = 0, if x is irrational or x = 0,
and 1/q, if x = p/q, (p,q) = 1 is a well-known example of a
function which is continuous on a dense set (the irrationals
and zero) and also discontinuous on a dense set (the nonzero
rationals). Another question which may be asked is: "Where
is f(x) differentiable?" Since f(x) = f(x+1) if x \= 0 and
x \= -1, it suffices to consider the interval 0 <= x <= 1.
Furthermore, since f is not continuous on the set of nonzero
rationals, the only possible points of differentiability are
the irrationals and zero. The object of this note is to give
an elementary proof that the function is not differentiable
at these points.
If x = 0 and h \= 0, [f(x+h) - f(x)]/h = f(h)/h. Let {h_i}
be a sequence of irrationals having zero as a limit. Then
lim(i --> oo) f(h_i)/h_i = 0. Now let h_i = 1/i, i = 1, 2, ...
Then f(h_i)/h_i = (1/i)/(1/i) = 1. Therefore lim(h --> 0)
f(h)/h does not exist and f is nondifferentiable at x = 0.
If x is an irrational number 0 < x < 1 and if h \= 0,
[f(x+h) - f(x)]/h = [f(x+h)]/h. If {h_i} is a sequence
of real numbers having zero as limit such that x + h_i
is irrational for each i, then lim(i --> oo)
[f(x + h_i)]/h_i = 0. Let the decimal representation of
x be .a_1a_2...a_n... Choose h_i = .a_1a_2...a_i - x.
Since x \= 0, a_i \= 0 for some i. Let N be the least
integer such that a_N \= 0. Then f(x + h_i) = f(a_1a_2...a_i)
>= 10^(-i) for all i >= N, and |h_i| <= 10^(-i). Hence,
| [f(x + h_i)]/h_i | >= 1 for all i >= N. Therefore
lim(h --> 0) [f(x+h)]/h does not exist and f is not
differentiable.
---------------------------------------------------------------
[13] Gerald Arthur Heuer, "Functions continuous at irrationals
and discontinuous at rationals", abstract of talk given
2 November 1963 at the annual fall meeting of the Minnesota
Section of the MAA, American Mathematical Monthly 71 #3
(March 1964), 349.
The complete text of the abstract follows, with minor
editing changes to accommodate ASCII format.
Earlier results of Porter, Fort, and others suggest additional
questions about the functions in the title. Differentiability
and Lipschitz conditions are considered. Special attention is
paid to the ruler function (f) and its powers. Sample results:
THEOREM: If 0 < r < 2, f^r is nowhere Lipschitzian; f^2 is nowhere
differentiable, but is Lipschitzian on a dense subset of the
reals. THEOREM: If r > 0, f^r is continuous but not Lipschitzian
at every Liouville number; if r > 2, f^r is differentiable at
every algebraic irrational. THEOREM: If g is continuous at
the irrationals and not continuous at the rationals, then
there exists a dense uncountable subset of the reals at
each point of which g fails to satisfy a Lipschitz condition.
REMARK BY RENFRO: The last theorem follows from the following
stronger and more general result. Let f:R --> R be such that
the sets of points at which f is continuous and discontinuous
are each dense in R. Let E be the set of points at which f
is continuous and where at least one of the four Dini derivates
of f is infinite. Then E is co-meager in R (i.e. the complement
of a first category set). This was proved in H. M. Sengupta
and B. K. Lahiri, "A note on derivatives of a function",
Bulletin of the Calcutta Mathematical Society 49 (1957),
189-191 [MR 20 #5257; Zbl 85.04502]. See also my note in
item [15] below.
---------------------------------------------------------------
[14] M. D. Mavinkurve, "Derivatives of functions discontinuous
on the rationals: Solution to Monthly Problem 5181",
American Mathematical Monthly 72 #3 (March 1965), 326-327.
The complete text of the item follows, with minor
editing changes to accommodate ASCII format.
5181 [1964, 325]. Proposed by E. J. Burr, University of New
England, Australia
Let f(x) = 0 for x irrational, f(0) = epsilon_1 > 0, and
f(m/n) = epsilon_n > 0 for m, n coprime integers with n > 0.
Find, or disprove the existence of, sequences {epsilon_n}
with lim(epsilon_n) = 0 such that (a) f'(x) exists nowhere,
(b) f'(x) exists for some x, (c) f'(x) exists for all
irrational x.
Solution by M. D. Mavinkurve, Siddharth College, Bombay,
India. For every sequence {epsilon_n} of the stated kind,
the function f will be continuous at irrational points
and discontinuous at rational points. f' can therefore
exist only at irrational points and, where it exists,
can have only the value zero.
(a) If we take epsilon_n = 1/n, the derivative will not
exist at any point. For if b be any irrational,
|b - m/n| < 1/[n^2 * sqrt(5)] eventually by Hurwitz's
theorem, and |f(b) - f(m/n)| / |b - m/n| > n*sqrt(5).
(b) If we take epsilon_n = 1/n^4, the derivative exists at
all quadratic irrationalities b. For then |b - m/n| > 1/n^3
eventually and |f(b) - f(m/n)| / |b - m/n| < 1/n --> 0.
(c) No sequence {epsilon_n} of the stated kind can ensure the
existence of f' at all irrational points. For by a theorem
due to M. K. Fort, the points of continuity of a real
function (on any interval) which itself is discontinuous
on a dense subset, form a set of the first category, and
the set of irrationals is not of the first category. For
if it were, together with the countable set of rational
numbers, the complete real line would be of the first
category.
Also solved by Eugene Allgower, J. O. Herzog, G. A. Heuer,
Solomon Marcus (Rumania), J. G. Mauldon (England), I. J.
Schoenberg, W. C. Waterhouse, and the proposer.
The substance of this problem has occurred in a variety
of papers. The earliest reference (furnished by Waterhouse)
is to W. D. A. Westfall, "A Class of Functions Having a
Peculiar Discontinuity", Bull. A.M.S., 15 (1909). Other
references containing solutions to this problem include:
M. K. Fort, Jr., "A theorem concerning functions
discontinuous on a dense set", this MONTHLY, 58 (1951).
I. J. Schoenberg, "The integrability of certain functions
and related summability methods", this MONTHLY, 66 (1959).
S. Marcus, "Sur les propriétés différentielles des fonctions
dont les points de continuité forment un ensemble frontiére
partout dense", Ann. Sc., École Norm. Sup. 79 (1962).
G. A. Heuer, "Functions continuous at irrationals and
discontinuous at rationals", (abstract), this MONTHLY,
71 (1964) 349.
---------------------------------------------------------------
[15] Gerald Arthur Heuer, "Functions continuous at the
irrationals and discontinuous at the rationals",
American Mathematical Monthly 72 #4 (April 1965), 370-373.
[MR 31 #3550; Zbl 131.29201]
Let f(x) = 0 if x is irrational, f(p/q) = |1/q| if
p and q are relatively prime integers, and f(0) = 1.
We say that a function g is Lipschitzian at x if there
exists a neighborhood U of x and a number M > 0 such
that |g(x) - g(y)| <= M*|x - y| for all y in U.
THEOREM 1: If 0 < r < 2, then the function f^r is nowhere
Lipschitzian. Moreover, f^2 is nowhere differentiable.
THEOREM 2: The function f^r is: (A) discontinuous at the
rationals for every r > 0; (B) continuous but
not Lipschitzian at the Liouville numbers, for
every r > 0; (C) differentiable at every irrational
algebraic number of degree <= r-1, if r > 3.
THEOREM 3: The function f^r is differentiable at every
algebraic irrational number if r > 2 (and, by
Theorem 1, at none if r <= 2).
THEOREM 4: The function f^2 is Lipschitzian but not
differentiable at the points of the set
{(1/2)*[m - sqrt(d)]: m is an integer
and there exists an integer n such that
d = m^2 - 4n is positive but not a perfect
square} . [This set is dense in the reals.]
THEOREM 5: If g is a function discontinuous at the
rationals and continuous at the irrationals,
then there is a dense uncountable subset
of the reals at each point of which g fails
to satisfy a Lipschitz condition.
(p. 373) "We omit the proof, because it is rather lengthy,
and one would hope to generalize the theorem by replacing
the rationals by an arbitrary dense set, and possibly to
show that the set of points at which g fails to be
Lipschitzian is a residual set."
NOTE: Sengupta/Lahiri had essentially obtained this result
in 1957 (the points of discontinuity have to form an
F_sigma set, however). See my remark in [13] above.
This result is also proved in Gerald Arthur Heuer,
"A property of functions discontinuous on a dense set",
American Mathematical Monthly 73 #4 (April 1966),
378-379 [MR 34 #2791]. Heuer proves that for each
0 < s <= 1 and for each f:R --> R such that
{x: f is continuous at x} is dense in R and
{x: f is not continuous at x} is dense in R,
the set of points where f does not satisfy a
pointwise Holder condition of order s is the
complement of a first category set (i.e. a co-meager
set). By choosing s < 1, we obtain a stronger version
of Sengupta/Lahiri's result. By intersecting the
co-meager sets for s = 1/2, 1/3, 1/4, ..., we get
a co-meager set G such that, for each x in G, f does
not satisfy a pointwise Holder condition at x for
any positive Holder exponent. (Heuer does not
explicitly state this last result.) A metric space
version of Heuer's result for an arbitrary given
pointwise modulus of continuity condition is essentially
given in: Edward Maurice Beesley, Anthony Perry Morse,
and Donald Chesley Pfaff, "Lipschitzian points",
American Mathematical Monthly 79 #6 (June/July 1972),
603-608 [MR 46 #304; Zbl 239.26004]. See also the last
theorem in Norton [17] below.
---------------------------------------------------------------
[16] James E. Nymann, "An application of Diophantine
approximation", American Mathematical Monthly 76 #6
(June/July 1969), 668-671. [MR 39 #4335; Zbl 177.07303]
Define f by f(x) = 0 if x is irrational and f(x) = 1/q
if x = p/q, where p and q are relatively prime with q > 0.
[Nymann overlooks defining f(x) for x = 0, but this has no
effect on his results.]
THEOREM 4: If 0 < r <= 2, then f^r is nowhere differentiable.
THEOREM 5: If r > 2, then f^r is differentiable at all
irrational algebraic numbers.
THEOREM 6: For each r > 0, there exist transcendental numbers
at which f^r is not differentiable.
THEOREM 7: Let h be any function from the positive integers
to the positive real numbers, and define f_h by
(f_h)(x) = 0 if x is irrational and
(f_h)(x) = 1/h(q) if if x = p/q, where p and q
are relatively prime with q > 0. Then there are
uncountably many transcendental numbers at which
f_h is not differentiable.
Note: Nymann's proof (which cites an existence result for
"stronger types" of Liouville numbers) actually shows
non-differentiability on a c-dense subset of R. Of
course, this follows from the stronger result that
the set is the complement of a first category set
(Fort). Indeed, the "stronger types" of Liouville
numbers themselves form a set whose complement is
first category (Marcus).
Marion K. Fort, "A theorem concerning functions discontinuous
on a dense set" American Mathematical Monthly 58 #6
(June/July 1951), 408-410. [MR1527895; Zbl 43.05503]
Solomon Marcus, "Les approximations diophantiennes et
la categorie de Baire", Mathematische Zeitschrift 76 (1961),
42-45. [MR 23 #A1598; Zbl 105.03804]
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D160327.html
http://tinyurl.com/y4p5g5
---------------------------------------------------------------
[17] Alec Norton [Kercheval], "Continued fractions and
differentiability of functions", American Mathematical
Monthly 95 #7 (Aug./Sept. 1988), 639-643.
[MR 89j:26009; Zbl 654.26006]
Define g:R --> R by g(x) = 0 if x is irrational, g(0) = 1,
and g(p/q) = 1/2^q if p and q are relatively prime with
q > 0.
PROPOSITION: There exists a partition A_0, A_1, A_2, ...
and A_oo of the irrational numbers, where each set is
c-dense in the reals, such that g is infinitely Peano
differentiable at each point of A_oo and, for each
n >= 0 and for each x in A_n, g is n-times Peano
differentiable but not (n+1)-times Peano differentiable
at x. Moreover, the complement of A_0 is a first
category set and the complement of A_oo is a Lebesgue
measure zero set.
NOTE: Norton says "uncountable dense sets" instead of
"c-dense in the reals". While it is a little
ambiguous what he means (uncountable sets that
are dense in the reals, or sets having an uncountable
intersection with every open interval) until one
gets to the proof, it is clear from the proof
(the sets involved are Borel, for instance)
that the sets are, in fact, c-dense in the reals.
Regarding Peano derivatives, this is easy to find on
the internet. Norton writes: "... the Peano derivative
agrees with the ordinary higher derivatives whenever
the latter is defined, and has the virtue of allowing
us to discuss higher derivatives in the context of a
dense set of discontinuities."
The complete text of Norton's remarks on p. 642 follow,
with minor editing changes to accommodate ASCII format.
Remarks. (1) The Proposition says that g is either not
differentiable at "most" points or infinitely differentiable
at "most" points, according to whether "most" is interpreted
in the sense of category or measure. This is related to the
well-known dichotomy between the Diophantine irrationals
and the Liouville irrationals (those which are not
Diophantine). See [Oxtoby's book] for more on this
interesting topic.
(2) Suppose we alter the definition of g so that 2^q
is replaced by w(q), where w:Z+ --> Z+ is some increasing
function. Then the following are left to the reader.
(See [Nymann's paper] for (a) and other related results.)
(a) If w(q) = q^2, then g is nowhere differentiable.
(Use (2).)
(b) If w(q) = q^3, then g is differentiable on a dense,
uncountable set of irrationals, but nowhere twice
differentiable.
(c) No matter how rapidly w increases, the set A_0
of points of nondifferentiability is residual.
As a consequence of (c), no function vanishing at the
irrationals and discontinuous at the rationals can be
differentiable at the irrationals. In fact, a little
more argument shows that no function can be discontinuous
at every rational but differentiable at every irrational.
(This last has been known, by another method of proof,
for some time, e.g. [Boas' "Primer of Real Functions"],
[Fort's paper].) The following theorem implies (c) and
the above statements, and provides a nice application
of the Diophantine approximation point of view. (A slightly
weaker version appears in [Heuer's 1966 paper] and is
considered from a more general viewpoint in [Beesley,
Morse, and Pfaff's 1972 paper].)
On p. 643, Norton proves the following result.
THEOREM: Let f:R --> R be discontinuous on a set of points
that is dense in R. Then there exists a co-meager
(i.e. residual) set B such that for all x in B
and for all s > 0, f fails to satisfy a pointwise
Holder condition of order (exponent) s at x.
NOTE: See also the comments I make in Heuer [15] and
Nymann [16] above.
---------------------------------------------------------------
[18] Richard Brian Darst and Gerald D. Taylor, "Differentiating
Powers of an Old Friend", American Mathematical Monthly
103 #5 (May 1996), 415-416. [MR1400724; Zbl 861.26002]
Define f:R --> R by f(x) = 0 if x is irrational or zero,
and f(p/q) = 1/q for p,q relatively prime with q > 0.
They note that the set of points at which f is not
continuous is the set of nonzero rational numbers.
THEOREM: If 1 < r <= 2, then f^r is differentiable only
at zero. If r > 2, then f^r is differentiable
almost everywhere (Lebesgue measure).
---------------------------------------------------------------
[19] Roger Alan Horn, "Editor's endnotes", American Mathematical
Monthly 104 #1 (January 1997), 92.
The relevant text of the note follows.
Harold Boas writes that the question of differentiability of
the ruler function discussed in a Note by Richard Darst and
Gerald Taylor in last May's issue [103 (1996) 415-416] has
a long history. Some previous MONTHLY papers that address
this problem are: Gerald J. Porter, On the differentiability
of a certain well-known function, 69 (1962) 142; G. A. Heuer,
Functions continuous at the irrationals and discontinuous
at the rationals, 72 (1965) 370-373; J. E. Nymann, An
application of Diophantine approximation, 76 (1969) 668-671;
Alec Norton, Continued fractions and differentiability of
functions, 95 (1988) 639-643.
---------------------------------------------------------------
[20] Marc Frantz, "Two functions whose powers make fractals",
American Mathematical Monthly 105 #7 (Aug./Sept. 1998),
609-617. [MR 99g:28018; Zbl 952.28006]
Following up on Darst/Taylor [18] above, Frantz investigates
the Hausdorff dimension of the graphs of f^r.
THEOREM 1: If r > 2, then the Hausdorff dimension of the
non-differentiability set for f^r is 2/r.
THEOREM 2: If r >= 2, then the set of points at which the
second order symmetric derivative of f^r doesn't
exist (finitely) is equal to the set of points
at which the ordinary derivative doesn't exist
for f^(r/2). In particular, this set has Hausdorff
dimension 4/r for r > 4.
The second order symmetric derivative of g at x = b
(a term Frantz doesn't use) is
lim(h --> 0) [ g(b+h) - 2*g(b) + g(b-h) ] / h^2.
The second half of Frantz's paper studies powers of
Cantor-like functions (non-decreasing functions that
are constant on the complementary intervals of various
Cantor sets).
---------------------------------------------------------------
[21] Roger Alan Horn, "Editor's endnotes", American Mathematical
Monthly 106 #3 (March 1999), 284.
The relevant text of the note follows, with minor
editing changes to accommodate ASCII format.
Gerald A. Heuer shared with us an email message to Marc Frantz
about the latter's article, Two Functions Whose Powers Make
Fractals, this MONTHLY 105 (1998) 618-630:
In your article in the August MONTHLY you refer to the
differentiability of the function (there called f)
sometimes called the "ruler" function, and make the
statement: "Darst and Taylor showed that if 1 <= r <= 2,
then f^r is nowhere differentiable, and if r > 2, then
f^r is differentiable almost everywhere." Darst and
Taylor did indeed show this in their 1996 MONTHLY article,
but the result is much older. G. J. Porter, this MONTHLY
69 (1962) 142, showed that f itself is nowhere differentiable,
and in the article Functions Continuous at the Irrationals
and Discontinuous at the Rationals, this MONTHLY 72
(1965) 370-373, some undergraduates and I proved that
if 0 < r < 2, then f^r is nowhere Lipschitzian (implying
nowhere differentiable), that f^2 is nowhere differentiable,
that for r > 2, f^r is almost everywhere differentiable,
and somewhat more. The results were extended further in
the note, A Property of Functions Continuous on a Dense
Set, this MONTHLY 73 (1966) 378-379.
At the time, I got some correspondence from Solomon
Marcus indicating that he had proved some related
results (I believe overlapping ours) earlier ... Of
course, as we both know, people are rediscovering old
results all the time, often without knowing of their
earlier establishment.
In his email response, Frantz wrote:
It looks as if the subject has an interesting history.
I find it particularly ironic that all the references
you mentioned were from the same journal! It makes me
think that someday there should be a computerized system,
more thorough and easy to use than anything available now,
which would help authors (and referees) avoid duplication
of results and give more thorough references. Until then
we'll have to settle for being educated after the fact.
Well, Marc (and all our authors, referees, and readers)
that day is almost here. If all goes well, sometime in 1999,
all of the MONTHLY from Vol. 1 in 1894 up to five years ago
(this cutoff rolls forward each year) will be available online
at JSTOR as graphic images and also in fully-searchable form.
Watch MAA Online for an announcement and details when this
exciting new service becomes available. Meanwhile, you can
learn more about the Andrew W. Mellon Foundation's Journal
Storage Project at www.mellon.org. If you visit www.jstor.org
and click on "Demo", you can experiment with searching and
viewing a demonstration database of three journals; clicking
on "About the Full JSTOR Collection" gives access to detailed
information about all aspects of JSTOR, including a list of
participating journals...it is a pleasure to note that the
MONTHLY is now among the latter.
---------------------------------------------------------------
[22] William Wade Dunham, "Nondifferentiability of the ruler
function", Mathematics Magazine 76 #2 (April 2003),
140-142.
Let f be defined on the open interval (0,1) by f(x) = 0
if x is irrational and f(x) = 1/q if x = p/q (lowest terms).
THEOREM: f is nowhere differentiable on (0,1).
The proof is elementary and does not make use of Diophantine
approximation results.
An ASCII version of this paper (that seems exactly
equivalent to the published version as far as I
can tell) is on the internet at
---------------------------------------------------------------
Dave L. Renfro
Valeriu Anisiu
> irrational, f(0) = 1, and f(x) = 1/q if x = p/q
> where p and q are relatively prime integers with q >
> 0.
>
> It is well-known that f is continuous at each
> irrational
> point and discontinuous at each rational point.
>
> On November 9, 2005 I posted some references about
> the
> differentiability of ruler-like functions.
>
> http://groups.google.com/group/sci.math/msg/cee496c760
> dd15be
>
> That post, although rather detailed for the topic,
> was hurriedly made. The present post is an extensive
> revision of that post.
>
> The ruler function itself seems to have been first
> introduced on pp. 14-15 of an 1875 book by Thomae:
>
> Carl Johannes Thomae, EINLEITUNG IN DIE THEORIE DER
> BESTIMMTEN INTEGRALE, Halle/Saale: Nebert, 1875.
> [JFM 7.0153.01]
>
> http://www.mathematik.uni-halle.de/history/thomae/inde
> x.html
> http://tinyurl.com/yxl78j [google translated
> version]
>
>
> Dave L. Renfro
>
It seems that the "Ruler Function" was first introduced
by B. Riemann (1826-1868) in his thesis.
Some papers refer to it as "Riemann's function".
V. Anisiu
Dave L. Renfro
> It seems that the "Ruler Function" was first introduced
> by B. Riemann (1826-1868) in his thesis.
> Some papers refer to it as "Riemann's function".
I looked through a copy of Riemann's 1854/1868 Habilitation
thesis and didn't see anything that appeared as if it was some
form of the ruler function. Although I can't read German at
all, the ruler function is something I think I would be able
to spot, if it was there. Riemann's paper is on the internet,
in case you or someone else wants to look. Maybe it's
there and I missed it.
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/
What you might have heard is that Riemann's paper is the
first time someone gave a function that is continuous on
a dense set and discontinuous on a dense set. However,
the function Riemann gave is defined by an infinite series
of functions, not as a piecewise defined function like the
ruler function. Of course, it might be that the function
Riemann gave is essentially equivalent to a ruler-like
function, I don't know right now. If this is the case,
I'd be interested to know it. Nonetheless, even if this
turns out to be the case, I think the simplicity of the
formulation that Thomae (apparently) came up with still
merits Thomae getting credit for the function as we know
it today.
If it's of any interest, I posted some information about
Riemann's work, and its impact on others, a few years ago:
http://groups.google.com/group/sci.math/msg/54110a92631f5625
(I note that where I wrote "...paper by Thomae in 1875",
it should have been "... book by Thomae in 1875".)
While I'm here, I should mention that (at least) one thing
in my post about the ruler function isn't correct. I wrote:
** For r > 2, f^r is differentiable on a set whose
intersection with every open interval has Hausdorff
dimension 1 - 2/r. Frantz [20]
This contradicts Darst/Taylor [18] (and probably some other
results I stated as well), who proved that if r > 2, then
f^r is differentiable almost everywhere (Lebesgue measure).
What Frantz [20] actually proved is that the set on which
f^r is nondifferentiable has Hausdorff dimension 2/r
(and it's trivial, by the way f is defined, that we must
therefore have f^r nondifferentiable on a set whose
intersection with every open interval has Hausdorff
dimension 2/r). This doesn't imply that the complement
has dimension 1 - 2/r, even for Borel sets. In the following
post, I gave an example of c-many pairwise disjoint Borel sets
each of which has positive Hausdorff dimension in every open
interval, which implies we can do this with a collection of
sets whose Hausdorff dimensions are bounded above zero.
[We can't have, for each n = 1, 2, 3, ..., fewer than c-many
of the sets in the original collection having Hausdorff
dimension less than 1/n, since a countable union of sets
of cardinality less than c has cardinality less than c.
(The cofinality of c is greater than omega.)] By the way,
I suspect we can actually have c-many pairwise disjoint
Borel sets of real numbers each with Hausdorff dimension 1,
but I don't know how to get this.
http://groups.google.com/group/sci.math/msg/8640af3ad9125d5b
Dave L. Renfro
Dave L. Renfro
[big snip]
I'm presently working on a very extensive expansion of
my December 13, 2006 post "Differentiability of the Ruler
Function". This present post is simply to keep the thread
active in google until I'm done, which should be at most two
to three weeks from now. [The original was 39.7 KB of ASCII,
and presently I'm up to 102.4 KB of ASCII, to give an idea
of what "very extensive expansion" means.]
Dave L. Renfro