半连续函数的性质与应用资料

(()PAGE\*ROMANPAGE\*ROMANII要.,.,,——半.从而得到了比更泛.通过对半研究,对半分析奠定了理基础.首先简述点讨半,保号到紧致空间半.,最终问题.实际上半理古典分析现代分析都着较泛.比如在最优化问题、变分等式问题、补问题及对策问题都着举足轻作.半；；AbstractCategoryoffunctionisverycomplicated.Characterizationandapplicationofcontinuousfunctionsareveryimportantinthefunctionakindoffunctionisalsocontinuous,itscharacterizationissimilarwiththecontinuousfunctions,whichiscalledextensionofthecontinuousfunctionssemi-continuousfunctions,thusakindoffunctionwithmorewindercharacterizationisobtained.Throughthehalfofthecontinuousfunctioninthemathematicalanalysiscontinuousfunctionwhichlayatheoreticalfoundationfortheapplication.First,thispaperexpoundsthenatureofthecontinuousfunctionandapplication,andthendiscussesthenatureofthesemi-continuousfunctions,detailedmathematicalandapplication,introducedthenumberoftopologicalspace,heapplicationofsemi-continuousfunctions.Finallydifferentiatecontinuousfunctionandsemi-continuousfunctionsproperties,application,andfinallyapplicationcontinuousfunctionsemi-continuousfunctionsnaturesolutionofproblem.Halfacontinuousfunctionintheclassicaltheoryanalysisandmodernanalysishasawiderangeofapplications.Forexample,inthemostproblems,variationalinequalities,phaseproblemsandcountermeasuresforthetheoryofandsoonallhasapivotalrole.Keywords: semi－continuous; continuous; functions;(()录III112223一致其427下半7运算7保号8无介值性 8界 8内有界 9保半10拓扑空半运算其确界其紧致空半长度半317半与比较 17半与别 184决问题 20结22参考文献 23致谢 24(()--PAGE21-绪 论.,.,...()证闭区间界存在存变量取值空间从维延拓般拓进行深入很者研究课.国内,、19996讨值Lipschitz,并值.,,,2008年2发表充条件主针对闭区间充进行构造阶梯国MagassyOUSMANE,WUCong-xin于20022模糊讨模糊、.、积及凸等很科均.关于,同集合同表述:献[1]距离空间半,献[2]Banach空间,献[3]给拓扑空间式可[4-14]分述,第二部再详细讲述础,证明闭区间界.给判闭区间充条件至情形也同样可仿照进行.拓扑空间介绍运算及确界给两.并且对紧致空间进行之后第三章主辨异同.运对比方法解决难点.终把际结合,把结合人日常生活践去,更好运书知识解决日常际.1函数,一.在人生活的自然界存在许有很的生长都地变化现象在关系上性．1.1[15性)1定义)

0,

0,使得,当

xx0

时,恒有则称在x 处．0

h(x)h(x)0mxx0

h(x)h(x0

),称hx ．0例1.1 易知h(x)x2在x0处,因为mx

h(x)mx0

(x2)2h(0)函数的性质函数的局部性质及应用如果f在x ,那么f在x 处有极限,且其极限值与值0 0相,根据极限性质能推断出f在U(x性态．0

f(x)0定理1.1局部有界性)如果h在x 处,那么h在某U(x)内有界．0 0证mxx0

h(x)A,取

1,则

0

xU(x0

;)有h(x)Ah在U(x;0

1

h(x)

A1定理1.2(局部保号性)f在x ,且0

f(x)00

f(x0

0),则对任何正r

f(x(或0

f(x),存在某U(x,使得对一切xU(x有0 0 0fx)r f(x)r)1在具体应用局部保号性r21

f(x0

f(x0

0时,在某U(x,0f(x)

f(x2

)成立．定理1.3(四则运算性)若g和h在x ,则gh0

gh

gh(这里h(x0

0)x0.1.4().fx处0,hu ,0

u f(x0

)h

x.0hu,0

0 0uu1 0

1h(u)h(u0

(1-1)u 0

f(x0

),u

f(x)x,0

0

0xx0xx0

,uu0

f(x)

f(x0

,(1-1):

0,

0,当,h

x0

h(f(x))h(f(x01.2 x1

x2)．x2g(u)cosuf(x)1x2

x2)x2))cos01x1上连续函数的基本性质1.5h[a,b]h[ab值小值．h[ab确界原到,h值域h([ab确界,记M．[ab]Mx[ab]h(xM成立,令1g(x ,x[ab道g[abg[abMh(x)假Gg一存在0g(x)

1Mh(x)

G,

x[a,b]h(x)

1,G

x[abMh([ab确界(小界)相矛盾,[ab)小值．

.h[ab值同到h[ab1.2界f[a,b],f[a,b]值．1.6()fa,b].

f(a)

f(b),u

f(a)f(b

f(a)u

f(bf(au

fb))x (a,b)0

f(x0

)u.1.3

0, n,一x ,得0nrxnr(x称rn次)即算,记作x nr0 0 0x时xn,因aanr

,因为f(x)

xn,a],

f(0)r

f(a)

,故由一点x (0,a)f(x)xnr.0 0 0一xxnr有1xnxn0

(x0

xxn11 0

xn20

xn1)01因(xn1

xn2

xn10

x 0即x x.0 0 1 1 0 1 1 0一致连续性及其应用义1.4 若h义I,对

0,

(0

x',

x''I,

x''有

h(x')h(x'')则称hI一致．1.7(一致)ha,b],h在a,b一致．1.4 I右端c1

cI,I 左端也c1 2

cI2

II 可1 2分别限无限:h分别II一致1 2,h

I 1

I 也一致．2

0,由hI和I 一致, 0,

0

x',1 2 1 2x''I,只要x'1

x''

有1h(x')h(x'')x',

x''I,只要x'1

x''

有2h(x')h(x'')xcI右端hc左I左端hc右1,所以hc,因对

0,3

0

2xc

时有3h(x)h(c)2,,)1 2 3

x',

x''I

x'x''对x'

x''

I1

I,h(x')h(x'')2

x'

x''

II,x'I1 2 1

x''I则2x'c

cx'

x''

3h(x)h(c)2

h(x')h(c)2

, h(x'')h(c) 而2h(xh(x'')hI一致连续.例题1.5 证:区间(a,b)有穷个一致连续的和它们的乘积此区间内仍是一致连续的．有穷个相加相乘可逐次解两个相加设h(x)

g(x)

都区间

(a,b)一致连续,

,h(x)

(ab一致连续, 0使(ab中1

xx

x''

1h(x')h(x''2又为g(x)(a,b)一致连续, 0,使(a,b)中,2有

x'x''

x''

,2g(x')g(x'')2,}当x'1 2

x''

（x'x为(ab [h(x')g(x')][h(x'')g(x'')]

h(x')h(x'')

g(x')g(x'')

2 2g(x)

h(x(ab是一致连续的．性质1.5 如果F()有限区间(a,b)是一致连续的,那么F()(a,b)必有界．证

0,

0(ab中

x',

x''

x''

F(x')F(x'')当a

a

ax''

a

F(x')F(x'')；当b

x'

b

b

x''

F(x')F(x'')

此,根据柯西收敛准,F(a0

xa0

F(xF(b0)

xb0

F(x例题1.6 闭区间[a,b]定义F():F(x)

axbF(F(a0)F(b0)

xaxbF()a,b,,F()(a,b)1.6 h()g()(a,b)一致，那么h()g()(a,b)也是一致证L0,

M0,使得h(x)L,

g(

(a

xb)0,根据h(xg(x(ab的一致,取0(ab中的x'x'',当x'x''时,有 h(x')h(x''得

,g(x')g(x''2M 2Lh(x')g(x')h(x'')g(x'')h(x')h(x'')g(x')h(x'')g(x')g(x'')] 得到h(x)

2M 2Lg(x(ab是一致的．2,,,.,.“”语言来叙述定义．定义2.1

0,

0,当

xx0

时,有h(x)h(x0

)

h(x)x处．00,

0

xx0

时,有h(x)h(x0

)

h(xx处0下．h(xx处h(xx处同时、0 0下．例2.1 假

f RRy

f (00x0y

f (x)y

xn所有偶整y,

f 点0下,所有奇整y,y

f 点0既不下y也不．上下的性质运算性质及应用2.1 如果闭区间a,b,g(),

h(x（下)们和g(x)h(x也闭区间（．22 如果闭区间a,b,h()和g()

0(或h(xg(x)

0,且下们g()h()闭区间a,b,如果h()

0,

g(x)

0）g(x)h(x)．2.3 如果闭区间a,b,g(间a,b下（)．

0（下）

1g(x)

闭区24 如果g(

x 处,而且有g(

)0,

0,使得x(x0

,x0

时有g(x)

0 00．如果g(x)

点x 处下,而且还0

g(x0

)

,那么

0,使得x(x0

,x0

g(x)

0．2.5 h()[a,b]上,上下)半连,那么有h(x)h(x[ab上有上(下)界,即MM．

0使得

x[ab时,有h(x[ab

xab,使得0h(x0

)sup{h(x):x[a,1．应用用半连续的定义证明2.1h(xg(x都是上半连续的,x[a,b],0

0,

0当

xx,0

x[abh(x)h(x) 0 2

g(x)

g(x)0 2所以h(x)

g(x)

h(x0

)g(x0

)

h(x)

g(x[ab2．应用上半连续的等价描述2.1g(xh(x[ab上上半连续,故

x[a,b]时0

limh(x)h(x),

g(x)

g(x)

xx0

0 xx 00xx0

(h(x)g(x))

h(x)xx0

limg(x)xx0h(x0

)g(x)0

h(x)

g(x)

[a,b]上上半连续．定理2.2(上半连续的局部保负)即是g()x处上半连续,0g(x0

)0,那么

0xx0

,x0

g(x)0.同理可得下半连续具有局部保正．介值性半连续定理不成立．g (g ()例22

0x1

12g(x

x12ag(1g(0没有x使得g(x)a．函数的界定理2.3 有界上的上半连续必有上界,并且能达到上确界.也就是:g(x[abg(x[abM

0,对

x[ab存

g(x

M．g(x

[a

x [a,0

,使得g(x

sup(g(x存．0x[a,b]证1.应用反证法,假g(x

[a,b]上无界,

x [a,b],使ng(xn

)n

(n{xn

中存收敛的子序列{x nk

,使得x xn 0k(当k

)．又因为[ab为x0

[a,b],但是g(xnk

)nk

,当k

时,g(x nk

g(x)xx

g(x

[a,b]上上半连续,g(x)xx0

g(x0

g(x0

0与题意相互矛盾．

g(x

supg(x)xE

g(x[ab上达不到上确界,那么

x[a,b],

g(x

M,

Mg(x)0,所以得到

1Mg(x)

[a,b]上上半连续,从而有上界,1

M'0,使得

x[ab

1Mg(x

M'因而得出g(x

M

,这与MM'

supg(x相互矛盾．xE2．假supx[a,b]

g(x)

g(x)M．0由supx[a,b]

g(x)得到,

x [a,b],使得M为g(x)点x 处的子极限,因为函0 0ggx0结．

limg(x)Mxx0

．推出M

g(x0

)．利用有限覆盖定理可以证明x[a,b],UOr xr

(xr

,xr

)g(x)r

g(xr

)

(xO)．r从[ab的开覆盖xr

|x [a,br

中可以造出有限个子覆盖xi

ni1

于是得到Mg(xi

)1|

,,,

f(x)[ab上有界性质2.3 g(x)(a,b)内上(或下)半连续,那么必然存内[[ab．使得g(x[]上保持有界．证g(x)(ab内下半连续,[]上无界,得到

[,]

[abg(x)1．

x(a,b)有g(x)1,因为g(x)下半连, 1使得1 1 1

xg(x)1

11．

[x1

,x1

]2

(a,b)g(x),1

xg(x2

)2又g(x),2

0,g(x)

2．

x2

[x2

,x2

]2 1 2

n

n

2 2n

(n) ,g(x)n．n,

n

(n

f()n

(n,f().一是2

f (x)y

xny增加递减f(x)

x时但f(x)]保半连续性

0,

0x时质2.4 假

f (x)E是义,是,那么就有yf (x)y

f(x)

(y,

xE有f(x)1

f(x)2

f (x)y

f (x)y1

f (x)y

f(x)

f()E1

x E0

f(x0

limn

f (xy

0

0nN,f (x)y 0

f(x0

)2y是固,

f (x)y

E,

0xExx0

(x)y

f(x0

)f

(x)y

f(x),

f(x)

f (x),f(xy

f(x0

出f(xE注记2.5 如果[a,b]

[ab义,那么y

f y

f (x)yf (x)y

f(x)f(x)

f(x)；f(x)2.6 的序列

f(x)[ab,,一个递减得y

f (x)y

f(x)．

f(x)1

f(x)2

f (x)y2.6从方逼近．证(构造

f (xxyy

y

xx的,已知

fx'是

f(x')

y

xx'的到[ab确界．即x[a,b]使得

f(x)

yx

f(x')x'[a,b]

y

x}令f ()x{f(x')yx'x}（证明f

(x)由式得到

x'[a,b],y a,b] yf (x)y

f(x)

y

x

f(x')

y

x得到f (x)y

f(x(x'))nf(x(x'))n

x(x')xnx(x')x'n

yx'xf (x')y

yx'x

(x')y

f (x)y

y

x.此时对于

x',

x[ab都成立,

x'x互换也同时成立,因而得出

f (x')y

f (x)y

yx'x表明

(x[ab．y要证得

(x)y

m

nf (y

f(x(x))m

x(x)xmf(x(x))mx(x)xf

(x)

(x)．y

m m m序列

(x下界的证明,x在yf (x)=y

f(x)

y

x

f(x')

yx'x中令x'

x,得到

f (x)y

f(x)

,故而

x[a,b],

{f y

g(x)

y

f (xy

(x)y

f(x)．g(x)

f(x)

,f(x)

,

0,

0,当x'[a,b],

x'x

f(x')

f(x)

f(x),[a,b],x,x,x(xn

xf (y

f[xy

(x)]

x(x)xyxy

x,x(x1

,x1

)xyk

(x),f(x)[a,b],M

0,f(x)

Mf (yk

f[xyk

(x)]

yxk yk

(x)xMnxy yk

(x)x

yk

f (x)y

g(xn

N0

,n

,xx(x)y

,f(x')

f(x)

到f(x(x))y

f(x)

f

(x)y

f(x(x))y

x(x)y

f(x(x))yf (x)y

f(x)

, n

, g(x)

f(x)

,

0

, 道g(x)

f(x)g(x)

f(x)．上半连续函数的性质,,,X,R,N,

u(x0

)x0,UX ,0

{x}n

n

xn

}X,,

AA2.7 X一个,g:

R,

x X．0g(x)x处,那么0

0Uu 0

xU,恒g(x

g(x0

)；g(x)x处下,那么0

0,Uu0

xU,恒g(x)

g(x0

)；g(x)x处,那么0

0,Uu0

xU有g(x)g(x；0g(x)（下）,那么X每一（下）g(x)x,g(x)x0 02.7 G,2.7 X,f

:XR,

xX0

①f

aR,

f1(aX（

f1(a,)X

aR,

f1[a,)闭于X

f1(,a]X)．其应用运算基本形式图谱广．定理2.8 X

g:

R,h:XR如果g(x；

h(x均、们g(x)

h(x()g(x(),那么g(x)；

0(1g(x)1g(x()g(x()2情况同理以证明12证明里就不予以证明．证 x 0

,

1f(x0

,因g(x))

x ,则0Uu0

xU

,恒g(x)

g(x)0

1

2(x)1g(x

0)因有g(x)

0,

00xU有01 g(x)

1g(x)1

g(x)0

10g2(x)1g(x)0

1 g(x)0由于xX0

,得到

1g(x)

0X例2.8[16]X

g:X

R,

h:XR如果g(x

h),均(),g(x)

h(x)

(x

X),存g(x)h(xX；如果g(x),g(x)

0,但h(x,h(x)

0,那么存g(x)h(xX；g(x)g(x)h(x)X

g(x)

0

h(x)

h(x)

01g()h()

0,g(x)

0,h(x)

0

x X,00g(x,

h(x)X,Uu(x)

xU有0g(x)

g(x0

),0

0h(x)h(x0

)0[g(x)[g(x)h(x)] (g(x)h(x))20 0 0 00 2g(x)h(x)(g(x0

)0

)(h(x0

))0

g(x0

)h(x0

)g(x)h(xX

x 0

,

0,

u(x0

),

xU有0g(x)h(x)h(x)g(x) (h(x)g(x))20 0 0 00

g(x0

),02

h(x0

)0

h(x)00g(x)h(x)(g(x0

)0

)(h(x0

))0

g(x0

)h(x0

)

g(x)

h(x)X

x 0

,

g(x0

)0

h(x)

0g(x)h(x)0, (01(g(x)h(x

)))0 0 2 0 0gh,Uu(x0

)

xU有g(x)h(x)h(x)g(x) (h(x)g(x))20 0 0 00

g(x0

) ,02

h(x)h(x0

)00其应用2.9[17S一族XR实值每一个hSX,那么

f(x)

inf{h(x)|hS}

XhSXg()suph()|hS}

X上

1([a,

{xX

f(x)a}

h1[a,

{x

:h(x)}h1([a,)){xhS

:h(x)a,hS}

xh1([a,)),0

hSh(x0

)af(x0

)inf{h(x0

)|hS}ax 0

f1[a,)x 0

f1([a,))f(x)0

af(x0

)inf{h(x0

)|hS}a

hSh(x0

)ax0

h1([a,hSg1([a,])f

h1([a,))hS

h

,

a

h1([a,))

X,到h1([a,))hS

f1([a,))X,f上的半连续函数R2.10 一个紧致空间ER

内下fE的af(axE

f(xmxE

f(x)

m得f(xx集合E非空

E族包含关系全序,这由E递增E交集不空集a,

m,f(a)

f(a另外,由m义,

m故f(a)m．推2.10 总一个紧致空间E(,]内下映射在E下界事实mf(a类似结如果我们应用这些结果,就重新原先取下确界和确界结下一节就研究这些结变分法一个重要应用．长度的半连续性一条曲线长度这条曲线；当曲线在我们就要明确意义下变动时,人们可能期待它长度也初等例子表明那样：C 方程yn1sinn

(0

xnN平面曲线.立这些曲线同样一个

当n

线段[0,].一致收敛不蕴含长度的收敛．可以修改用任何于等于的数代替l小于的数代替l.[0,]的曲线长度的下极限等于正是下半连续性,我们精确的表述事实．参数化曲线空间:TR的一E是一所有从TE内的连续映射定义一条参数化曲线,于是可以考虑从TE内的连续映射的集合(TE作为T上的Ed(f,g)supd(fg(t))tT定义的一致收敛的与(TE上的拓扑．

(TELff定义在拓扑空间(T,E)上的数值函数．定理211 度L(f)是T,E)的f证 对于T的所有有限子集

t,tt

,对于所有f(TE

1 2 n

1 2 nV (f)d(f(tii

f(t i1对于所有aT,从(TEEff(a是连续的,于是对于所有,映射fV

(f)是连续的．R推211 从T,E)到射f (f差)R对于变分法的应用：单变量变分法的问题直译是在给定的曲线集合里求一条曲线,其长度是最小的．3与比较.,,3.1 g()[a,b](),g()[a,b]()g(x)[a,b],,一列{x[abg(xn

)

(n1,2,),因为{xn

}为,必收敛子{x }．nkxk

A因为axnk

b

A[ab,从而g(x)A点

1 00

x0

x[abg(x)

g(A)1又因为limxk

A对于0

0,k

xnk

0

kK时有n g(x )k nk

g(A)

(kK)即n k

g(A)

(k

),

[ab同理g(x)[ab]().进而出()不仅3.2 g(

[a,

,x0

[ab使得g(x)0

supa,b]

g(x)．g(x)

[a,

A,即对

x[a,b],Ag(x)0,令G(x)

1 ,Ag(x)

x[ab.因为g(x[a,b],

0,

0

xx0

g(x)

g(x0

)

x[ab)，即01A1G(x)

a

1 G(x)0

1 G(x)

1G(x0

1G(x)

为,因而G(x),以G(x)[ab],B为G(x)一个,有0G(x)

1Ag(x)

B

1g(x)A1B

(x[a,b])A

sup

g(x),.[a,b]与区别,3.3 f(x)1

x0x0 x0

1g(x)1

x0x0x0

f(x),g(x),

f(x)

g(xx03.4 3qq

xq

q

14 q

xq

qf(x)23

xq

q

g(4 q

xq

q

13 q

xq

qh( 3 q

xq

q

p)qf(xg(x、h(x都,24 q

xq

qf(g(x)h(x)24

xq

q

0

[a,b]y,则yy

f (x)y

f(x)

f(x)

[ab

y,y

f (x)y

f(x)

f(x[ab.

f(x),x[a,b]

x

0f(x)f(x)

limf

0 n(x)n n

0 y y 0y

n

f(x)n

f (xy

)

f (x)y n

f(xn

f (x)y n

f(x)n

f (x)y 0

f (x)y n

f (x)y 0

(.y

f(x[ab,

f (x)[a,b]yy,

f (x)y

f(x),

f(x)