On the critical case of Okamoto's continuous non-differentiable functions

On the critical case of Okamoto's continuous non‑differentiable functions

著者 Kobayashi Kenta

journal or

publication title

Proceedings of the Japan Academy Series A:

Mathematical Sciences

volume 85

number 8

page range 101‑104

year 2009‑01‑01

URL http://hdl.handle.net/2297/37862

doi: 10.3792/pjaa.85.101

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 every- where’ 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 non- dierentiable almost everywhere at the critical case.

Key words: Continuous non-dierentiable function; the law of the iterated logarithm.

1. Introduction. We consider a parameter- ized family of continuous functions which were pre- sented 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 ffng¹_n¼0 of piecewise linear and contin- uous functions. For a xed parametera2 ð0;1Þ, each function in the sequence is dened as follows:

(i) f₀ðxÞ ¼x;

(ii) f_nþ1ðxÞis continuous on [0,1], (iii) f_nþ1 k

3ⁿ ¼f_n k 3ⁿ ; f_nþ1 3kþ1

3^nþ1

¼fn

k 3ⁿ þa f_n kþ1

3ⁿ

f_n k 3ⁿ

; f_nþ1 3kþ2

3^nþ1

¼fn

k 3ⁿ þ ð1aÞ fn

kþ1 3ⁿ

fn

k 3ⁿ

; f_nþ1 kþ1

3ⁿ

¼fn

kþ1 3ⁿ

;

fork¼0;1; ;3ⁿ1;

(iv) fnþ1ðxÞis linear in each subinterval k

3^nþ1xkþ1 3^nþ1 for

k¼0;1; ;3^nþ11:

Figure 1 shows the operation from fn to fnþ1. Oka- moto’s functionF_aðxÞis then dened as

FaðxÞ ¼ lim

n!1fnðxÞ:

He noticed thatF_aðxÞis continuous on½0;1and co- incides with some known functions whenatakes par- ticular values. For example, the cases a¼5=6 and a¼2=3 correspond to nowhere-dierentiable func- tions dened by Perkins [5] and Bourbaki [1] respec- tively. Also, if a¼1=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 thatF_aðxÞhas the following fea- tures:

(i) If a < a0, then FaðxÞ is dierentiable almost everywhere.

(ii) If a0< a <2=3, thenFaðxÞis non-dierentiable almost everywhere.

(iii) If 2=3a <1, then F_aðxÞis nowhere dieren- tiable.

Here, the constanta0ð¼0:5592 Þis the unique real root of

54a³27a²¼1:

2000 Mathematics Subject Classication. Primary 26A27;

Secondary 26A30.

doi: 10.3792/pjaa.85.101 62009 The Japan Academy

As for the case a¼a0, it remained open whether F_aðxÞ is dierentiable almost everywhere or non- dierentiable almost everywhere. In this case, we proved thatFaðxÞis non-dierentiable almost every- where.

3. Main result. The main result of this arti- cle is the following

Theorem 1. If a¼a₀, then F_aðxÞ is non- dierentiable almost everywhere in½0;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 x¼X¹

n¼1

nðxÞ

3ⁿ ; nðxÞ 2 f0;1;2g;

denote the ternary expansion of x2 ½0;1Þ. If x is a rational number of the formk=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:

cðkÞ ¼ 1; ðk¼0 or k¼2Þ;

2; ðk¼1Þ;

and

SnðxÞ ¼Xⁿ

k¼1

c kðxÞ

; TnðxÞ ¼1

nþ1ðxÞ ¼1

SnðxÞ;

where 1ðAÞ is the indicator function that takes the value one if argumentAis true and zero otherwise.

With these denitions, we have the following lemma:

Lemma 1.

lim sup

n!1

TnðxÞ ffiffiffin p 1 holds for almost every x2 ½0;1Þ.

Proof.Since thecðnÞare 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

Fig. 1. The operation fromfntofnþ1. Before the operation (top) and after the operation (bottom). This operation is performed in each subinterval½k=3ⁿ;ðkþ1Þ=3ⁿ.

Fig. 2. The graph of Perkins’s function (top), Bourbaki’s function (middle) and the Cantor-Lebesgue singular function (bottom).

lim sup

n!1

SnðxÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4nlog logn p ¼1 and

lim inf

n!1

SnðxÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4nlog logn p ¼ 1 almost everywhere inð0;1Þ.

Thus in particular, the events S_nðxÞ=pffiffiffin 1 and SnðxÞ=pffiffiffin

1 both happen innitely often.

Each time SnðxÞ=pffiffiffin

exits the interval ½1;1Þ, it must do so with a value of

c

_nþ1ðxÞ

¼ 2 (the only negative value). Thus,

T_nðxÞ ffiffiffin p 1

happens innitely often as well. r We now complete the proof of the main theo- rem.

Proof of theorem 1. We rst note thatFaðxÞ has the following representation:

F_aðxÞ ¼X¹

k¼1

_kðxÞ;

_kðxÞ ¼Y^k1

l¼1

p _lðxÞ

q _kðxÞ

; where

pð0Þ ¼a; pð1Þ ¼12a; pð2Þ ¼a;

qð0Þ ¼0; qð1Þ ¼a; qð2Þ ¼1a:

In what follows, we assume that a¼a0 and x satises

lim sup

n!1

TnðxÞ ffiffiffin p 1:

From the denition ofTnðxÞ, we can take an increas- ing sequencefrngwhich satises

X^rn

k¼1

c _kðxÞ

pffiffiffiffiffir_n

; _rnþ1ðxÞ ¼1; n¼1;2;3; : Here we denefxngby

xn¼X^rn

k¼1

kðxÞ 3^k : Then,

xx_n¼ X¹

k¼rnþ1

kðxÞ 3^k 1

3^rnþ1 >0

and we have the following evaluation:

F_aðxÞ F_aðxnÞ xxn

¼

P1

k¼rnþ1kðxÞ

P1

k¼r_nþ1kðxÞ=3^k

P1

k¼rnþ1_kðxÞ

1=3^rn

¼3^rnY^rn

l¼1

p lðxÞ

q

r_nþ1ðxÞ

þ X¹

k¼r_nþ2

Y

k1

l¼r_nþ1

p lðxÞ

q kðxÞ

3^rnY^rn

l¼1

p lðxÞ

q

r_nþ1ðxÞ

X¹

k¼rnþ2

Y

k1

l¼rnþ1

p _lðxÞ

q

_kðxÞ!

3^rnY^rn

l¼1

p lðxÞ

qð1Þ X¹

k¼1

pð1Þmax

0l2

pðlÞ^k1max

0l2qðlÞ

!

¼3^rnY^rn

l¼1

p lðxÞ

a ð2a1ÞX¹

k¼1

a^k

!

¼að23aÞ

1a exp X^rn

l¼1

log 3p lðxÞ

! : Using the following relations:

log

3pð0Þ¼log

3pð2Þ¼logð3aÞ;

log

3pð1Þ¼log

3ð12aÞ¼log27a²54a³ 9a²

¼log 1 9a²

¼ 2 logð3aÞ;

we obtain

F_aðxÞ F_aðxnÞ xxn

að23aÞ

1a exp logð3aÞX^rn

l¼1

c

lðxÞ! að23aÞ

1a ð3aÞ^pffiffiffiffi_rn : It follows that

n!1lim

FaðxÞ FaðxnÞ xxn

¼ 1:

Namely, F_aðxÞ is non-dierentiable at x. From the previous lemma, we know that

lim sup

n!1

TnðxÞ ffiffiffin p 1

holds almost everywhere in½0;1Þ, and so, we can con- clude that FaðxÞ is non-dierentiable almost every- where in½0;1Þ.

Acknowledgments. I would like to express my gratitude to Prof. H. Okamoto for informing me of his interesting study about continuous non- dierentiable functions and for encouraging me to write this article after I proved the case a¼a0. 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.

References

[ 1 ] N. Bourbaki,Functions of a real variable, Trans- lated from the 1976 French original by Philip Spain, Springer, Berlin, 2004.

[ 2 ] P. Hartman and A. Wintner, On the law of the iterated logarithm, Amer. J. Math. 63 (1941), no. 1, 169{176.

[ 3 ] H. Okamoto, A remark on continuous, nowhere dierentiable functions, Proc. Japan Acad. Ser.

A Math. Sci.81(2005), no. 3, 47{50.

[ 4 ] H. Okamoto and M. Wunsch, A geometric con- struction of continuous, strictly increasing sin- gular functions, Proc. Japan Acad. Ser. A Math. Sci.83(2007), no. 7, 114{118.

[ 5 ] F. W. Perkins, An Elementary Example of a Continuous Non-Dierentiable Function, Amer.

Math. Monthly34(1927), no. 9, 476{478.