C i g i f u n k c y j n e
N i e c h X z b i r ,
Y , d p r z e s t r z e D m e t r y c z n a .
" n "! f : X Y ,
o r a z n i e c h
n
f n "!.
c z y l i o k r e [l o n y j e s t c i g f u n k c y j n y
n
N i e c h f : X Y .
D e f i n i c j a
f n "! z m i e r z a p u n k t o w o d o f u n k c j i n a z b i o r z e X ,
M w i m y , |e c i g f u n k c y j n y f
n
f x n "! z m i e r z a d o f x
j e [l i d l a k a |d e g o x " X c i g w p r z e s t r z e n i Y
n
z m e t r y k d , t z n .
X
f f : ! " x " X l i m d f x , f x = 0 !
n n
n "
! " x " X " 0 "n 0 "! " n n 0 d f x , f x
n
F u n k c j f n a z y w a m y f u n k c j g r a n i c z n .
D e f i n i c j a
f n "! z m i e r z a j e d n o s t a j n i e d o n a z b i o r z e , g d y :
C i g f X
n
X
f
f : ! " 0 "n 0 "! " n n 0 " x " X d f x , f x
n
n
N i e c h
X , Y p r z e s t r z e n i e m e t r y c z n e ,
f : X Y ,
n
f : X Y .
D e f i n i c j a
f n "! z m i e r z a n i e m a l j e d n o s t a j n i e d o f u n k c j i f n a z b i o r z e X , g d y :
C i g
n
E
C o m p X
f
f : ! " E "C o m p X f
f
n
n
U w a g a
f n "! - z b i e |n y j e d n o s t a j n i e ! f n "! - z b i e |n y n i e m a l j e d n o s t a j n i e
! f n "! -
n n n
z b i e |n y p u n k t o w o .
- 1 -
P r z y k Ba d
N i e c h X = [ 0 ; 1 ]
o r a z n i e c h f x = x n d l a n "!.
n
y f ( x ) = x
1 . 2
y
f ( x ) = x ^ 2
f ( x ) = x ^ 3
1
f ( x ) = x ^ 4
f ( x ) = x ^ 5
f ( x ) = 0 . 1 5
0 . 8
f ( x ) = - 0 . 1 5
S e r i e s 1
0 . 6
f
1
f
2
f
3
1 f
0 . 4
f
5
f
2
f
f f 5
f 3 4
4
0 . 2
x
x
- 1 . 2 - 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 2 0 . 4 0 . 6 0 . 8
1 1 1 . 2
-
-
-
- 0 . 2
- 0 . 4
- 0 . 6
- 0 . 8
P o n i e w a |
- 1
0 d l a x "[ 0 , 1 )
l i m x n =
n " {
1 d l a x = 1
- 1 . 2
f n "! j e s t z b i e |n y p u n k t o w o . N a t o m i a s t n i e j e s t z b i e |n y j e d n o s t a j n i e , b o d l a
w i c c i g
n
x "[ 0 , 1 ) f
w y k r e s |a d n e j z f u n k c j i n i e z n a j d u j e s i w c a Bo [c i w p a s i e
n
[ 0 , 1 ) - , .
f n "!.
Z b a d a j m y t e r a z w X = [ 0 , 1 ) z b i e |n o [ n i e m a l j e d n o s t a j n c i g u f u n k c y j n e g o
n
N i e c h E " C o m p X . S t d n a p o d s t a w i e t w i e r d z e n i a c h a r a k t e r y z u j c e g o z b i o r y z w a r t e w
p r z e s t r z e n i s t a n d a r d o w e j , z b i r E j e s t d o m k n i t y i o g r a n i c z o n y , t z n .
E = [ a ; b ] , g d z i e 0 d"a d"b 1 .
P o n i e w a | o r a z z a t e m z t w i e r d z e n i a o 3 c i g a c h w y n i k a , |e
0 d"d x n , 0 d"b n 1 l i m b n = 0
n "
E
f
a s t d w y n i k a , |e f .
l i m d x n , 0 = 0 ,
n
n "
f n "! j e s t z b i e |n y n i e m a l j e d n o s t a j n i e w
Z a t e m X = [ 0 , 1 ) .
n
- 2 -
y f ( x ) = x
y
f ( x ) = x ^ 2
1 . 5
f ( x ) = x ^ 3
f ( x ) = x ^ 4
f ( x ) = x ^ 5
S e r i e s 1
S e r i e s 2
1
0 . 5
a b
x
- 0 . 4 - 0 . 2 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8
E
b
a
x
-
- 0 . 5
T w i e r d z e n i e ( o c i g Bo [c i f u n k c j i g r a n i c z n e j )
" n "! f "C X
n
! f "C X
X
}
f f
n
W n i o s e k
J e [l i f u n k c j a g r a n i c z n a c i g u f u n k c y j n e g o f u n k c j i c i g By c h n i e j e s t c i g Ba , t o c i g n i e j e s t
z b i e |n y j e d n o s t a j n i e .
- 3 -
P r z y k Ba d c . d .
0 d l a x "[ 0 ; 1 ,
J e |e l i f x = x n , X = [ 0 ; 1 ] , t o f x =
n
{
1 d l a x = 1 ,
f "C X
a w t e d y i f "C X
n
z a t e m
X
X
f
f .
n
T w i e r d z e n i e
f n "!"B ( X , Y ) , g d z i e B ( X , Y ) = { f | f : X Y , f - f u n k c j a g r a n i c z n a } .
N i e c h
n
W t e d y
X
f ! l i m d s u p f , f = 0
f n
n
n "
P r z y k Ba d
2
f : ! !
Z b a d a z b i e |n o [ j e d n o s t a j n c i g u , g d z i e f x = n x e - n x
n
n
D p D j
o r a z o k r e [l i o b s z a r y z b i e |n o [c i p u n k t o w e j i j e d n o s t a j n e j .
x " !.
I . A b y z b a d a z b i e |n o [ p u n k t o w , w y b i e r z m y
W t e d y
2
0 , g d y x = 0 ,
l i m f x = l i m n x e - n x =
n
{
n " n "
0 , g d y x `"0 ,
b o d l a x `"0 m a m y :
n x H x 1
"
l i m = = l i m = l i m = 0 .
2 [ ] 2 2
"
n " n "
e n x x 2 e n x n " x e n x
Z a t e m
" x " ! l i m f x = 0 ! f a"0 ! D p = !.
n
n "
f
I I . S p r a w d z a m y , c z y f .
n
f "C
n
f - x = - f x , t z n . f j e s t f u n k c j n i e p a r z y s t
n n
! f "B ! , !
n
n x H n
}
l i m f x = l i m = l i m = 0
2 2
n
x " x " x "
e n x 2 n x e n x
D l a d o w o l n e g o n "! m a m y
d s u p f , f = s u p #" f x - f x #"= s u p #" f x #"= s u p f x
n n n n
x " ! x " ! x "[ 0 ; " )
- 4 -
f ,
A b y w y z n a c z y k r e s g r n y w a r t o [c i f u n k c j i z b a d a j m y j e j e k s t r e m a l o k a l n e .
n
P o n i e w a |
2 2 2
f ' x = n e - n x - 2 n 2 x 2 e - n x = n 1 - 2 n x 2 e - n x = 0
n
1 - 2 n x 2 = 0
1
x =
2 n
m i n m a x
m i n m a x
x
x
1
1
1 1
-
-
2 n
2 n
2 2 n n
z a t e m
1 n
f m a x = f = .
n n
2 n 2 e
S t d
n n
s u p #" f x #"= m a x { f m a x , l i m f x } = m a x , 0 = ,
n n n
{ }
2 e 2 e
x " ! x "
w i c
n
f f
l i m d s u p f , f = l i m = "`"0 !
n
n
2 e
n " n "
W y k a |e m y , |e
D j = ( - " ; a ] *"[ b ; ") , g d z i e a 0 b .
- 5 -
y f ( x ) = x e ^ ( - x ^ 2 )
y
y
f ( x ) = 2 x e ^ ( - 2 x ^ 2 )
f ( x ) = 3 x e ^ ( - 3 x ^ 2 )
2
1 . 5
1
f
3
f
f 3
1
f 2
0 . 5
f
f 1 2
f 2
f
3
x x
x
- 2 - 1 . 5 - 1 - 0 . 5 0 . 5 1 1 . 5 2
- 0 . 5
- 1
- 1 . 5
- 2
P o n i e w a |
"n b : " n n b 1 b
1 2 n
" 0 !
n
{
2 n
1
"n a : " n n a - a
2 n
n > m a x { n a , n b }
N i e c h .
W t e d y
2 2
d s u p f , f = m a x { f b , | f a | , 0 } = m a x { n b e - n b , - n a e - n a } .
n n n
S t d
l i m d s u p f , f = 0 .
n
n "
Z a t e m
D j = ( - " ; a ] *"[ b ; ") , g d z i e a 0 b .
T w i e r d z e n i e ( o p r z e j [c i u d o g r a n i c y p r z y r |n i c z k o w a n i u c i g u f u n k c y j n e g o )
N i e c h p r z e d z i a B I " !
f : I ! , n "!
n
! l i m f ' x = f x
f "D I " n "!
n n
n
[ l i m ] '
n " n "
f n "!- z b i e |n y p u n k t o w o n a I
n }
f ' n "!- z b i e |n y j e d n o s t a j n i e n a I
n
- 6 -
T w i e r d z e n i e ( o p r z e j [c i u d o g r a n i c y p r z y c a Bk o w a n i u c i g u f u n k c y j n e g o )
N i e c h p r z e d z i a B I " !
f : I ! , n "!
n x x
x 0 "I
! " x "I l i m f t d t =
+" +"
n n
l i m f t d t
n "
x 0 x 0 n "
f - c a Bk o w a l n a " n "!
n }
f n "!- z b i e |n y j e d n o s t a j n i e
n
P r z y k Ba d c . d .
2
J e [l i f x = n x e - n x , t o f u n k c j a g r a n i c z n a f a"0 .
n
x 0 = 0 .
N i e c h W t e d y
x x
+"
n
l i m f t d t = +"0 d t = 0
n "
0 0
f
! f
n
x x
x
2 2 2
1 1 1
}
f t d t = n t e - n t d t = - e - n t = 1 - e - n x n "
+" +"
n
2 2 2
0
0 0
- 7 -