Real Analysis with Economic Applications - Figures docx

xm is divergent but it does not diverge to ∞... the unique fixed point ¯ x.f is not a contraction, and it has no fixed points... f is lower semicontinuous but not upper semicontinuous at

x− ε

M(xm) converges to x (xm) diverges to ∞ (xm) is divergent but

it does not diverge to ∞.

b b b

b b

b b

b b

b b

b b

b b

sin x

π 3 π 2

B

1 1

δ y

Nδ,X(y)

b

the unique fixed point ¯ x.

f is not a contraction, and

it has no fixed points.

p (x)

y

x S

x x x

b b

f is upper semicontinuous but

not lower semicontinuous at x.

f is lower semicontinuous but not upper semicontinuous at x f nor lower semicontinuous at x. is neither upper semicontinuous

1 2

1 4

1 2

3

1 4

1 8

3 8

5 8

7 8

b

bΦ(x)

B2

x1 x2 x1 x2 x1 x2

Not upper hemicontinuous at x1 and x2

Not lower hemicontinuous at x1 and x2

Not lower hemicontinuous at x1

Not lower hemicontinuous at x2

Upper hemicontinuous

Not upper hemicontinuous at x1

Not upper hemicontinuous at x2

Lower hemicontinuous

ω(A, B)

d H (A, B)

d H (A, B) ω(A, B)

A

B

A

B

S is convex S is not convex.

S is not convex.

S is not convex.

0 a base for C 0 not a base

for C

0 not a basefor C

y y

y

x

y

an algebraicinterior point of S

not an algebraicinterior point of SS

S− x∗

Y

xz

0is an algebraic interiorpoint of S, but it is not aninterior point of S

x− αe

S

1

2x −32e2x

Z

0

yx

x− y

x1x2= 1(1, 1) b

A

B

C

b αx

V

O

b

b b

b

b

Ywz

z− w

α

β(z − w)y

kx − yk.

This hyperplane equals

K−1

(sup K(S)) butK(x − z) 6= kKk∗

kx − zk

x1

x2