A C T A U N I V E R S I T A T I S L O D Z I E N S I S FO LIA M ATHEM ATICA 7, 1995
Grażyna Horbaczewska
R E M A R K O N S O M E T H E O R E M O F Z A J i Ć E K
For a ty p ica l co n tin u o u s fu n ctio n / on [0,1], / has an I-essential d e riv e d n u m b e r a t each p o in t x 6 (0 ,1 ).
Zajicek proved [3] th a t, for a typical continuous real-valued func tio n / an d each x €E (0,1), th ere exists y £ which is a n essential derived n u m b er of / at, x. In this p ap er we shall prove th a t th is th eo rem rem ains tru e if we replace th e notion of an essential derived n u m b er of / a t x by an analogous notion for th e B aire category.
Let C denote th e set of continuous real valued functions defined on [0,1] furnished w ith th e m etric of uniform convergence. W hen we say a typical / e C has a certain p ro p erty V , we shall m ean th a t th e set of / 6 C w ith this p ro p erty is residual in C.
T h e n o ta tio n used th ro u g h o u t this p ap er is stan d a rd . In p a rtic u la r, iH stan d s for th e set of real num bers, = 5H U { -o o ,o o } , I for th e cr-ideal of sets of th e first category, ||/ || for the no rm in C, B ( f , r ) for th e open ball in C w ith centre f and radius r an d \ A f° r th e ch aracteristic function of a set A.
D e f in itio n 1 . ([1]) We say th a t xo £ 5H is an u p p er /-d e n sity p o in t of a set A having th e Baire p ro p erty if and only if th e re exists an increasing sequence of real num bers {i„}ng;v tending to infinity, such
th a t
X i„ (B -i0)n ( -i,i) — * 1 w ith re sp e ctto I as n ooon ( - 1 , 1 ) (see [2] for th e definition of th e convergence w ith respect to I) . We shall use th e n o tatio n d j ( E , x o) = 1.
O bserve th a t xo is an u p p er I - density po in t of a set A if an d only if 0 is an u p p e r I - density point of A — xo = {x — xo : x £ A} .
It is easy to see th a t d j ( E , x o ) = 1 if and only if th ere exists an increasing sequence of real num bers {¿„}ngN tending to infinity, such th a t
n —*oo v
/ - a.e. on (-1,1).
D e f in itio n 2. We say th a t y is an /-essen tial derived n u m b er of / a t x if th ere exists a set E C having th e Baire property, such th a t d i ( E , x ) = 1 an d lim t^ x , t e E - - •¡I"- 5-- =
V-T h e o r e m . For a typical f £ C and each x £ (0 ,1 ), there exists y £ iH which is an I-essential derived n u m ber o f f at x.
Proof. Let {Pk}k£N be a sequence of polynom ials which is dense in C. For each k £ N , p u t M* = ||P ^ || = su p l 6 [01] \Pk (x)| a n d choose
6k such th a t 0 < 6k < ( k M k) ~ 1 an d 6k \ 0 as k —> oo. Let
g = n u b ( p ‘ - = n m r p B i P t ' •
m—1 k—m
For each m £ N , the set B ( P k , j f c ) is open and dense, so G is a dense G ^-subset of C. Hence G is residual in C. Choose an a rb itra ry / £ G. It is sufficient to prove th a t, for each x £ (0 ,1 ), th ere exists an /-e sse n tia l derived num ber of / a t x. Fix xo £ (0 ,1 ). Since f £ G, we can choose an increasing sequence of positive integers {fcn }n£jv such th a t / £ B ( P k n ,6kn • (4fc£)- 1 ) for each n £ N . Let h n = 6kn , A n = P'k (xo), z n = (A;n )- 1 . Since 6n \ 0 as n —> oo, we have h n \ 0 for n —* oo. F o r n large enough, we get (xq— h n , xq + h n ) C (0 ,1 ). We
R E M A R K ON S O M E T H E O R E M O F Z A Jl'C E K 27
shall show th a t, for such an n and for x e (x0 - h n , x0 - h n( k n ) ~ l ) U (a.’o + h n( k n ) ~ l , x o + /i„), we have
f i x ) - f ( x 0) x - x0 - A n
We can assum e th a t x <E (x0 + j ^ , x0 + h n ) (th e o th e r case is analogous). We have
(1)
f ( x ) - f i x o) P kn(X) - Pkn( x 0) x — Xo X — Xo < ~ A „ ( x ) | - f | / ( x 0) - P fc„(x0)| | x - X o | < « I ’ 1 1 « M M -1 2 k u -By th e T aylor form ula, for some f £ ( 0 ,1),p kn {x) - Pkni x 0) X — Xo - p kni x o) \p 'k n( 0 ( * - * o ) (2) 1 ‘ikfi From (1) an d (2) we ob tain f j x ) - / ( x 0) X — Xq - n > o ) < A- ’ hence (3) f i x ) ~ f i x o ) X — X o A n z n
D enote by y G iH a cluster po in t of a sequence { A n } n e N - T h en th ere exists a subsequence {np}peiv ° f th e sequence of positive in te gers, such th a t A nr —» y as p —* oo. Define
£ = 0 +
We shall show th a t
f i x ) — f ( x o ) d j ( E , x 0) = 1 an d lim ---=
y-A ccording to th e above rem arks, it is sufficient to show th a t th ere exists a sequence of real num bers {t „}n€N tending to infinity, such th a t
hm Xt n l E- z0)n ( -i,i)(* ) = X (-i,i)(® ) n —♦oo
I - a.e. on ( — 1,1 ). Let t p = ( h Up) 1. T hen
t p ■ ( E - x 0) D t T hr
' h n ”
" ( h knr ) U ( k , ’ O '
Since h np \ 0 for p —» oo and k n /* oo for n —► oo, we have t p / oo as p -*• oo and 1 / k ni \ 0 as / -> oo. Hence, for x £ (0 ,1 ), x ^ 0, th ere exists p0 such th a t, for each p > po,
1 6 ( ■ 1, _ ^ ) u ( s ; ’1) '
T h enx e t,
h " ”
c tp ■ ( E - Xo),
so, for p > po, we have
1-R E M A 1-R K ON S O M E T H E O 1-R E M O F Z A jfĆ E K 29
T herefore d j ( E , x o) = 1. We only need to show th a t
• f j x ) - f j x o) _ lim
M e0 x ~ x °
Let e > 0. T h ere exists p0 such th a t 1 / k „ p < e /2 for p > p 0. T h en , for each x £ E such th a t \x — x 0| < h npo, we have
* € (J
°° r ' h Up, xi 0 — -— I h n, \ U ( l xo + -— , xq + h nh „p-p o kn,: kn.
From (3) it follows th a t there exists p > p0 such th a t f j x ) - f j x q)
X — Xq
- A n , 1 £
< Zn” ~ JT~ K ^ np *2‘
Since A nf —* y as p —> oo, there exists p\ such th a t, for p > we get
\A *r
Let p2 = ma x { p o , p i ) and 6 = ftnp2- According to th e above rem arks, for x G E , if \x — Xq| < S, then we have
f j x ) - f i x o)
a; — xq - y < £. T his com pletes th e proof.
Re f e r e n c e s
[1] T . F ilip czak , On som e abstract d en sity topologies, Real A nalysis E x ch an g e 14 (1988-89), 140-166.
[2] W . P o re d a, E. W agner-B ojakow ska an d W W ilczyński, A category analogue o f the d en sity topology, F und. M ath . 1 2 5 (1985), 167-173.
[3] L. Z ajicek, O n essen tia l derived n um bers o f typical c o n tin u o u s fu n c tio n s , T a tr a M o u n ta in s M ath . P u b l. 2 (1993), 123-125.
G. H O R B A C Z E W SK A
G rażyna Horbaczewska
U W A G A N A T E M A T P E W N E G O T W I E R D Z E N I A L. Z A iĆ K A
Zajićek ([3]) udowodnił, że typow a funkcja rzeczyw ista m a w każ dym punkcie x £ (0 ,1 ) istotną, liczbę pochodną. W pracy tej dowo dzimy, że tw ierdzenie to pozostaje prawdziwe, jeśli zastąpim y p o ję cie istotnej liczby pochodnej przez analogiczne pojęcie d la kategorii B aire’a.
In s titu te of M a th e m a tic s Łódź U n iv ersity ul. B an ach a 22, 90 - 238 Łódź, P o la n d
