Internat. J. Math. & Math. Sci.
(1986) 517-519
517
Vol. 9 No. 3
A SIMULTANEOUS SOLUTION TO
TWO PROBLEMS ON DERIVATIVES
F.S. CATER
Department of Mathematics
Portland State University
Portland, Oregon 97207
(Received March 7, 1985)
Let A be a nonvoid countable subset of the unit interval [0,I] and let B be
an F -subset of [0 17 disjoint from A
Then there exists a derivative f on [0 I] such
that 0fl, f
0 on A, f>O on B, and such that the extended real valued function i/f
ABSTRACT.
is also a derivative.
KEY WORDS AND PHRASES. Derivative, primitive, Lebesgue summable, knot point.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 26A24
In this note, we construct a derivative f such that I/f is also a derivative, and
f and i/f have some curious properties mentioned in [I] and [2].
(By an Fo-set in the
real line, we mean the union of countably many closed subsets of R.)
THEOREM i.
of [0,i] disjoint from A.
f
We prove
Let A be a nonvoid countable subset of [0,I] and let B be an F -subset
Then there exists a measurable function f on [0,I] such that
0 on A, f > 0 on B, 0f[ on [0,1] and
(I)
f is everywhere the derivative of its primitive,
(2)
i/f is Lebesgue summable on [0,I],
I/f is everywhere the derivative of its primitive.
(3)
Here we let
I/0.
and A is dense, we will obtain a simple example of a derivative that
When m(B)
vanishes on a dense set of measure 0.
From [2] we infer that there exists a derivative f vanishing on A and positive on
B.
From [i] we infer that there exists a derivative g infinite on A and finite on B.
However, Theorem
prove Theorem
provides a simultaneous solution to both of these
problems.
Finally, we use these methods to construct a concrete example of functions
g2
To
we will employ some of the methods used in [3].
gl
and
that have finite or infinite derivatives at each point, such that the Dini
derivatives of their difference,
In all that follows, let
gl-g 2,
(n(i))i=
satisfy certain pathological properties.
denote the sequence of integers I, I, 2, I, 2,
3, i, 2, 3, 4, i, 2, 3, 4, 5,
Proof of Theorem i.
Let
(ai)i=
be a sequence of points in A such that each point
of A occurs at least once in the sequence.
that
A is a finite set.)
(Here we do not exclude the possibility
We assume, without loss of generality, that B is nonvoid.
F.S. CATER
518
Let
BICB2CB3c...cB i
be an expanding sequence of closed sets such that B
t,i=l Bi"
(Here we do not exclude the possibility that B is a closed set.) Let u i denote the
-1/2
As in [3] we put (x)
(I + Ixl)
to the set B
distance from the point a
n(i)
i
For each index j, put
gj(x)
+
’=I (2iul(x-an(i)
g(x)
+
i=l (2iul(x-an(i) ))’
i for which a
for aeA, because there are infinitely many indices
Then g(a)
i/-.
Here we let 0
I/g(x).
f(x)
I/gj(x),
f.j(x)
))’
forb e B; note that if b e B k, then
On the other hand, g(b) <
an(i).
(2 k) < 2
(2ku l(b-an(k)))
i:k (2iul(b-an(i) 11
-1/2k,
i <
We infer from Lemma 4 of [3], that g is Lebesgue summable on [0,i].
g(x)-gj(x)
and since g>l,
gj>l,
h-
limh+ 0
Take any e>0.
gj
. g-g-’J
it follows that
ix+h
x
>
f’-f3
g(x).
g(t)dt
/x+h
x
lhl
g(t)dt
].x+h
gj(t)dt
x
lh-I /X+hx
f’j(t)dt
> 0.
By Lemma 4 of [3], we have
Select an index j so large that
are continuous, when
lh-i
ih-1
> 0,
g(x)gj(x)(fj(x)-f(x))
Now choose any x with g(x) <
Note also that
f.(x)-f(x)<g(x)-gj(x)<e.
Since f. and
is small enough we have
< e,
g(x)
gj(x)
< e,
fj(x)
< e.
For such j and h we obtain
h
-I
x+h
x+h
I
x
(g(t)-gj(t))dt
<
g(xl-gj(x)
+
g(tldt
g(x)[
gj(t)dt
gj(x)
3e.
fj(x)
f(x)
f
x
+
From 0 < f o-f
lh-
lh-lfx
x+h
g-g. we obtain
[h-i fx_x+h
f(t)dt-
It follows that lim
h+O
[h-I
f(x)[
fx+h
x
f (t)dt-
fj(x)
+
x+h
(f (t)- f(t))dt
+ h -1 I x
h-I /x+h
x
-1
-<_ 2e
+
=<
-I
+ h
f(t)dt
2e
h
f(x).
fX+hx (fj(t)
/X+hx
(g(t)
f(t))dt
gj (t))dt
< 5e.
519
TWO PROBLEMS ON DERIVATIVES
=.
Choose any x with g(x)
Since
t,
gj
Take any N>O.
gj(x)>N.
gj(t) > N. For
Select j so large that
It-xl
< d implies
is continuous, there is a d>0 such that
-I
< d,
for
It follows that
and
such
lhl
f(t)<fj(t)<N
g(t)>gj(t)>N
-I x+h
g(t)dt >N O<h f
h-i fx+h
x
f(t)dt <N
X
Finally,
h-i yx+h
x
g(t)dt
h/0
lim
h/0
h-I /x+h
x
f(t)dt
lim
g(x).
f(x).
0
This completes the proof
When m(B)
[0,1]\Bas
in
i, we do not know if our argument can be modified to make f
[2]
0 on
Perhaps this requires an approach altogether different from ours
We say that x is a knot point of the function F if its Dini derivatives satisfy
D+F(x)
D-F(x)
and
D+F(x)
D_F(x)
-.
We conclude by presenting a simple and direct example of functions gl and g2 having
derivatives (finite or infinite) at every point such that gl-g has knot points in
2
every interval (Consult [4] for analogous examples)
Let
{ai}i=
and
{bi}i=
be countable dense subsets of (0,I) that are disjoint
(c(t-d))dt for c>0, d>0, x>0.
Let Z(c,d,x)
2c-l[(l+cd)
(l+cd-cx)
{ 2c-l[(l+cd)
Z(c,d,x)
=I z(2iu ’an(i
for 0<x<l.
2]
to the set
b 1,
x)
ai}.
Put
g2(x)
(R)
finite on A.
B, and
g2’
g’
on A and
{x:
Put g
g’
D+g(x)
on B.
=}, E 2
-} is a dense
D_g(x)
gl-g2.
with
x d,
if x > d.
bi}
and let v
i
denote the
i -I
v
Ai=1 Z(2 i ,b n (i) ,x)
By the argument in the proof of Theorem
absolutely continuous functions on (0,1)
E
1/2]
if
+ (l+cx-cd)-
Let u. denote the distance from a
n(i)
i
to the set {a
distance from b
n(i)
gl(x)
We integrate to obtain
gl
we prove that
on A
g’2
gl and g2 are
gl finite
on B
Then g is absolutely continuous on (0,i),
Each of the sets
D-g(x)
{x:
G-subset
many open dense subsets of (0,i).
=}, E 3
{x:
D+g(x)
{x:
-} and E
4
of (0,i), i.e., is the intersection of countably
It follows that
EIOE20E3nE4
is also a dense
G-subset
But each point in this intersection is a knot point of g, even though
of
(0,1).
g2
have derivatives (finite or
on
gl and
infinite) everywhere by the proof of Theorem I.
REFERENCES
I.
Application des Proprits Descriptives de la Functions Contingente
la Gomtrie Diffrentielle des Vari@t@s Cartsiennes, J. Math. Pures et
CHOQUET, G.
App,. 26 (1947), 115-226.
2.
3.
4.
ZAHORSKI, Z. Sur la Premiere Drive, Trans. Amer. Math. Soc. 69 (1950), 1-54,
(Lemma 11).
KATZNELSON, Y. and STROMBERG, K. Everywhere Differentiable, Nowhere Monotone
Functions, Amer. Math. Monthly, 81 (1974), 349-354.
BELNA, C.L., CARGO, G.T., EVANS, M.J. and HUMKE, P.D. Analogues to the DenjoyYoung-Saks Theorem, Trans. Amer. Math. Soc. 271 (1982), 253-260.