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8/13/2019 beamerAnalysis6-13
1/14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
50
100X = [0,1] i.e., X compact ; f discontinuous
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
50
100X = (0,1] i.e., X not compact; f continuous
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
50
100X = [0,1] i.e., X compact; f continuous
Figure 1. Compactness plus continuity implies boundedness
() August 18, 2013 1 / 14
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(
(
X
Y
W
X
Y
[ )
[
[xx xxxn xn
) f(x)
n)
f(xn)
S
f1(S)
O
f1(O)
if part of the proof (contra-positive)
f continuous, f1(S) not open impliesSnot open
only if part of the proof (contra-positive)
fdiscontinuous,O open,f1(O) not open
Figure 1. fis continuous iffSopen = f1(S) open
() August 18, 2013 2 / 14
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Continuity & Hemi-continuity defns: continuity
1(a) Inverse image formulation of continuity.
A functionf : X Y is calledcontinuousif for every open setO Y,f1(O)is an open set ofX.
1(b) Neighborhood formulation of continuity.
A functionf : X Y is calledcontinuousif for everyx Xand everyneighborhoodUoff(x)there is a neighborhoodV ofxsuch thatf(x) U foreveryx V.
1(c) Sequential formulation of continuity.
A functionf : XRn is calledcontinuousatx0 Xif whenever {xm}m=1convergesx0 then {f(xm)} converges tof(x0); the functionfis continuous if itis continuous atxfor everyx X.
() August 18, 2013 3 / 14
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Continuity & Hemi-continuity defns: upper hemi-continuity
2(a) Inverse image formulation of upper hemicontinuity.A correspondence : S T is calledupper hemicontinuousif for every open
setO T, the upper inverse image ofOunder,1
(O) S, is an open set.Explosions allowed;implosions are not
2(b) Neighborhood formulation of upper hemicontinuity.A correspondence : S T is calledupper hemicontinuousif for everys Sand every neighborhoodUof(s)there is a neighborhoodV ofssuch that(s) Ufor everys V.
2(c) Sequential formulation of upper hemicontinuity.
A compact-valued correspondence : S T is calledupper hemicontinuousif for everys S, every sequence {sn} converging tosand every sequence{tn} withtn (sn), there is a convergent subsequence {tnk} of {tn} such thatlim
ntnk=
t (
s)
.
() August 18, 2013 4 / 14
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Continuity & Hemi-continuity defns: lower hemi-continuity
3(a) Inverse image formulation of lower hemicontinuity.A correspondence : S T is calledlower hemicontinuousif for every opensetO T, the lower inverse image ofOunder,1(O) S, is an open set.Implosions allowed;explosions are not
3(b) Neighborhood formulation of lower hemicontinuity.
A correspondence : S T is calledlower hemicontinuousif for everys S,and every open setU T withU
(s) = /0, there exists a neighborhoodV of
ssuch thatU(z) = /0, for everyz V.
3(c) Sequential formulation of lower hemicontinuity.
A correspondence : S T is calledlower hemicontinuousif for everys S,anyt (s), and any sequence {sn} converging tos, there exists a sequence{tn} such thattn (sn)and lim
ntn= t.
() August 18, 2013 5 / 14
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Explosions, Implosions and Hemi-continuity
Thegraphof a correspondence:
Graph() = {(s, t) ST :t (s)}
Theouter accumulationof a correspondence.
OAc
(s) = {t T : (s,
t)is an accumulation point of Graph()}Theinner accumulationof a correspondence.
IAc(s) = {t T : {sn} s,{tn} s.t.n, tn (sn)and {tn} t}
Definition: explodesat sif(s) IAc(s).
Definition: implodesat sifOAc(s) (s).
Theorem: If:S T,Tis compact and is compact-valued, then is u.h.c atsiff does not implode ats
Theorem: is l.h.c atsiff does not explode at s() August 18, 2013 6 / 14
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O)
(
( )( )1(O)
1(O)
Figure 3. Upper and Lower inverse images of
() August 18, 2013 7 / 14
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(1,
2)
(0,0)
x
u
is
uhsb
ut
not
lhc
u1((1, 2)) = {0} closed
u1
((1, 2)) =
(1
,1)
(0,0)
is
lhc
but
not
uhc
1
((1, 1)) = {0} closed
1((1, 1)) = R
Figure 4. u and
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0 1
U
O
yz w
(
(
)
)
x
Figure 5. : [0, 1] R() August 18, 2013 9 / 14
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0 1
U
O
yz w
(
(
)
)
v x
Figure 6. : [0, 1] R() August 18, 2013 10 / 14
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Berges Theorem
Berges Theorem of the Maximum: If :X Yis a continuous
correspondence with nonempty and compact values and:YR is acontinuous function, theny :X Ydefined byy(x) =argmaxy(x)(y)isu.h.c. and :XR defined by(x) =maxy(x)(y)is a continuousfunction.
For our purposes, think of:Xas a space of price vectors
Yas a space of commodity vectors
as a budget correspondence, continuous, compact-valued
as a utility function, continuousy as a demand correspondence. Result: its u.h.c.
as an indirect utility function. Result: its continuous
(Ill adduto the pics, its a direct utility function)
() August 18, 2013 11 / 14
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Berges theorem: Introduction
0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 01 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 1 0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 01 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 10 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 2.5
* **
y1y1y1
y2y2y2
y(1)
y(2)
y(0.5) (1) (2)(0.5)
P = x1x2
Figure 7. The demand correspondence is u.h.c
() August 18, 2013 12 / 14
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Berges theorem: Role of upper-hemi-continuity
(
(
}(y) = (x)y O (nbd of(x))inf(O)
U =1(O)
(x)
(x)
x
x
V = 1(U)
RX Y
Figure 1. Lower hemi-continuity of implies that (x)>inf(O).
() August 18, 2013 13 / 14
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Berges theorem: Role of upper-hemi-continuity
Y
(
(
(
(
}
}(y) = (x) O (nbd of(x))
Uh =1(O)
O (nbd of((x))
O
=O O
sup(O)
(x)
x
Vh =1(Uh)
RX
Figure 2. Upper hemi-continuity of implies that (x)< sup(O).
() August 18, 2013 14 / 14
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