No. 8]
Proc. Japan Acad., 85, Ser. A (2009)
101
On the critical case of Okamoto’s continuous
non-dierentiable functions
By Kenta KOBAYASHI
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University,
Kakuma-machi, Kanazawa, Ishikawa 920-1192, Japan
(Communicated by Masaki KASHIWARA, M.J.A., Sept. 14, 2009)
Abstract: In a recent paper in this Proceedings, H. Okamoto presented a parameterized
family of continuous functions which contains Bourbaki’s and Perkins’s nowhere dierentiable
functions as well as the Cantor-Lebesgue singular function. He showed that the function changes
it’s dierentiability from ‘dierentiable almost everywhere’ to ‘non-dierentiable almost everywhere’ at a certain parameter value. However, dierentiability of the function at the critical
parameter value remained unknown. For this problem, we prove that the function is nondierentiable almost everywhere at the critical case.
Key words:
Continuous non-dierentiable function; the law of the iterated logarithm.
1. Introduction. We consider a parameterized family of continuous functions which were presented by H. Okamoto [3, 4]. This function can be
regarded as a generalization of Bourbaki’s [1] and
Perkins’s [5] nowhere dierentiable functions as well
as of the Cantor-Lebesgue singular function.
Okamoto’s function is constructed as the limit
of a sequence ffn g1
n¼0 of piecewise linear and continuous functions. For a xed parameter a 2 ð0; 1Þ, each
function in the sequence is dened as follows:
(i) f0 ðxÞ ¼ x;
(ii) fnþ1 ðxÞ is continuous on [0,1],
k
k
(iii) fnþ1 n ¼ fn n ;
3
3
3k þ 1
k
¼
f
fnþ1
n
nþ1
3
3n
kþ1
k
þ a fn
f
;
n
n
3
3n
3k þ 2
k
fnþ1
¼
f
n
3nþ1
3n
kþ1
k
þ ð1 aÞ fn
fn n ;
3n
3
kþ1
kþ1
fnþ1
¼
f
;
n
3n
3n
for k ¼ 0; 1; ; 3n 1;
2000 Mathematics Subject Classication. Primary 26A27;
Secondary 26A30.
doi: 10.3792/pjaa.85.101
62009 The Japan Academy
(iv) fnþ1 ðxÞ is linear in each subinterval
k
kþ1
x nþ1
3nþ1
3
for
k ¼ 0; 1; ; 3nþ1 1:
Figure 1 shows the operation from fn to fnþ1 . Okamoto’s function Fa ðxÞ is then dened as
Fa ðxÞ ¼ lim fn ðxÞ:
n!1
He noticed that Fa ðxÞ is continuous on ½0; 1 and coincides with some known functions when a takes particular values. For example, the cases a ¼ 5=6 and
a ¼ 2=3 correspond to nowhere-dierentiable functions dened by Perkins [5] and Bourbaki [1] respectively. Also, if a ¼ 1=2, Fa is the Cantor-Lebesgue
singular function which is non-decreasing and has
zero derivative almost everywhere (Fig. 2).
2. Dierentiability of Fa . In the paper [3],
H. Okamoto proved that Fa ðxÞ has the following features:
(i) If a < a0 , then Fa ðxÞ is dierentiable almost
everywhere.
(ii) If a0 < a < 2=3, then Fa ðxÞ is non-dierentiable
almost everywhere.
(iii) If 2=3 a < 1, then Fa ðxÞ is nowhere dierentiable.
Here, the constant a0 ð¼ 0:5592 Þ is the unique real
root of
54a3 27a2 ¼ 1:
102
K. KOBAYASHI
[Vol. 85(A),
Fig. 1. The operation from fn to fnþ1 . Before the operation (top)
and after the operation (bottom). This operation is performed in
each subinterval ½k=3n ; ðk þ 1Þ=3n .
As for the case a ¼ a0 , it remained open whether
Fa ðxÞ is dierentiable almost everywhere or nondierentiable almost everywhere. In this case, we
proved that Fa ðxÞ is non-dierentiable almost everywhere.
3. Main result. The main result of this article is the following
Theorem 1. If a ¼ a0 , then Fa ðxÞ is nondierentiable almost everywhere in ½0; 1Þ.
In order to prove this theorem, we need some
denitions and a preliminary lemma concerning with
the law of the iterated logarithm [2].
Denitions. Let
x¼
1
X
n ðxÞ
;
3n
n¼1
n ðxÞ 2 f0; 1; 2g;
denote the ternary expansion of x 2 ½0; 1Þ. If x is a
rational number of the form k=3n , we use the ternary
expansion ending in all 0’s (instead of the one ending
in all 2’s). We also use the following notations:
1; ðk ¼ 0 or k ¼ 2Þ;
cðkÞ ¼
2; ðk ¼ 1Þ;
Fig. 2. The graph of Perkins’s function (top), Bourbaki’s function
(middle) and the Cantor-Lebesgue singular function (bottom).
and
Sn ðxÞ ¼
n
X
c k ðxÞ ;
k¼1
Tn ðxÞ ¼ 1 nþ1 ðxÞ ¼ 1 Sn ðxÞ;
where 1ðAÞ is the indicator function that takes the
value one if argument A is true and zero otherwise.
With these denitions, we have the following
lemma:
Lemma 1.
Tn ðxÞ
lim sup pffiffiffi 1
n
n!1
holds for almost every x 2 ½0; 1Þ.
Proof. Since the cðn Þ are i.i.d. random variables
with mean 0 and variance 2 with respect to Lebesgue
measure on ð0; 1Þ, the law of the iterated logarithm
[2] implies that
No. 8]
On the critical case of Okamoto’s continuous non-dierentiable functions
Sn ðxÞ
lim sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1
n!1
4n log log n
and
Sn ðxÞ
lim inf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1
n!1
4n log log n
almost everywhere in ð0; 1Þ.
pffiffiffi
Thus in particular, the events Sn ðxÞ= n 1
pffiffiffi
and Sn ðxÞ= n 1 both happen innitely often.
pffiffiffi
Each time Sn ðxÞ= n exits the interval ½1; 1Þ, it
must do so with a value of
c nþ1 ðxÞ ¼ 2
(the only negative value). Thus,
and we have the following evaluation:
P
1
k¼rn þ1 k ðxÞ
Fa ðxÞ Fa ðxn Þ
¼ P1
xx
ðxÞ=3k
n
1=3rn
rn Y
¼ 3 rn
p l ðxÞ q rn þ1 ðxÞ
l¼1
1
X
3 rn
rn Y
p l ðxÞ l¼1
1 k1
X
qð1Þ max qðlÞ
pð1Þ
max pðlÞ
k1 Y
p l ðxÞ q k ðxÞ ;
where
pð0Þ ¼ a;
pð1Þ ¼ 1 2a;
qð0Þ ¼ 0;
qð1Þ ¼ a;
pð2Þ ¼ a;
qð2Þ ¼ 1 a:
In what follows, we assume that a ¼ a0 and x
satises
Tn ðxÞ
lim sup pffiffiffi 1:
n
n!1
From the denition of Tn ðxÞ, we can take an increasing sequence frn g which satises
rn pffiffiffiffiffi
X
c k ðxÞ rn ; rn þ1 ðxÞ ¼ 1; n ¼ 1; 2; 3; :
k¼1
Here we dene fxn g by
rn
X
k ðxÞ
:
3k
k¼1
Then,
1
X
k ðxÞ
1
r þ1 > 0
k
n
3
3
k¼r þ1
!
0l2
rn 1
Y
X
¼3
ak
p l ðxÞ a ð2a 1Þ
!
rn
l¼1
¼
n
0l2
k¼1
k ðxÞ;
xn ¼
!
k1 Y
p l ðxÞ q k ðxÞ
k¼rn þ2 l¼rn þ1
l¼1
x xn ¼
rn Y
p l ðxÞ q rn þ1 ðxÞ
1
X
p l ðxÞ q k ðxÞ þ1
k1
Y
k¼rn þ2 l¼rn
l¼1
k¼1
k ðxÞ ¼
1
X
þ
Tn ðxÞ
pffiffiffi 1
n
Fa ðxÞ ¼
k¼rn þ1 k
P
1
k¼rn þ1 k ðxÞ
3 rn
happens innitely often as well.
r
We now complete the proof of the main theorem.
Proof of theorem 1. We rst note that Fa ðxÞ
has the following representation:
103
k¼1
að2 3aÞ
exp
1a
rn
X
l¼1
!
log
3p l ðxÞ :
Using the following relations:
log 3pð0Þ
¼ log 3pð2Þ
¼ logð3aÞ;
27a2 54a3 log 3pð1Þ
¼ log 3ð1 2aÞ
¼ log
9a2
1 ¼ log
2 ¼ 2 logð3aÞ;
9a
we obtain
Fa ðxÞ Fa ðxn Þ
xx
n
rn X
að2 3aÞ
exp logð3aÞ
c l ðxÞ
1a
l¼1
pffiffiffiffi
að2 3aÞ
ð3aÞ rn :
1a
It follows that
Fa ðxÞ Fa ðxn Þ
¼ 1:
lim n!1
xx
n
!
104
K. KOBAYASHI
Namely, Fa ðxÞ is non-dierentiable at x. From the
previous lemma, we know that
[ 1
Tn ðxÞ
lim sup pffiffiffi 1
n
n!1
[ 2
holds almost everywhere in ½0; 1Þ, and so, we can conclude that Fa ðxÞ is non-dierentiable almost everywhere in ½0; 1Þ.
Acknowledgments. I would like to express
my gratitude to Prof. H. Okamoto for informing me
of his interesting study about continuous nondierentiable functions and for encouraging me to
write this article after I proved the case a ¼ a0 . I
would also like to thank an anonymous referee for
letting me know a much shorter proof of Lemma 1
using the law of the iterated logarithm.
[Vol. 85(A),
References
[ 3
[ 4
[ 5
] N. Bourbaki, Functions of a real variable, Translated from the 1976 French original by Philip
Spain, Springer, Berlin, 2004.
] P. Hartman and A. Wintner, On the law of the
iterated logarithm, Amer. J. Math. 63 (1941),
no. 1, 169{176.
] H. Okamoto, A remark on continuous, nowhere
dierentiable functions, Proc. Japan Acad. Ser.
A Math. Sci. 81 (2005), no. 3, 47{50.
] H. Okamoto and M. Wunsch, A geometric construction of continuous, strictly increasing singular functions, Proc. Japan Acad. Ser. A
Math. Sci. 83 (2007), no. 7, 114{118.
] F. W. Perkins, An Elementary Example of a
Continuous Non-Dierentiable Function, Amer.
Math. Monthly 34 (1927), no. 9, 476{478.