Real Analysis with Economic Applications - Figures docx

x + ε x x − ε M (x m ) converges to x. (x m ) diverges to ∞. (x m ) is divergent but it does not diverge to ∞. x 1 x 2 x 1 x 2 x 1 x 2 Figure A.1 x + ε x x − ε y x M lim sup x m = x lim sup x m = y and lim inf x m = x lim sup x m = ∞ and lim inf x m = x x 1 x 2 x x 1 x 2 x 1 x 2 Figure A.2 sin x cos x π 3π 2 1 −1 ln x 1 e x 1 Figure A.3 ➊ ➋ ➌ ➍ ➎ ➏ ➐ ➑ ➒ A × B A B Figure B.1 45 ◦ line 45 ◦ line 1 1 1 1 f f Figure B.2 1 −1 1 −1 C ∞ C 3 C 2 C 1 Figure C.1 x ε δ y N δ,X (y) N ε,X (x) Figure C.2 f 3 f 4 f 6 1 1 1 6 1 4 1 3 1 2 Figure C.3 f x f(x) f 2 (x) x f(x) ¯x 45 ◦ line 45 ◦ line f f is a contraction with the unique fixed point ¯x. f is not a contraction, and it has no fi x ed points. Figure C.4 45 ◦ line f g := f − id [0,1] x 1 1 Figure D.1 [...]... upper semicontinuous nor lower semicontinuous at x Figure D.3 1 f1 1 2 f2 1 4 f3 1 2 1 1 4 1 2 3 4 Figure D.4 1 1 8 3 8 5 8 7 8 1 f +ε fm f f −ε 0 1 Figure D.5 1 f1 f2 f3 f4 1 Figure D.6 1 7 8 f3 1 2 f2 -1 f1 1 Figure D.7 r(x) x Φ(x) 0 B2 Figure D.8 Γ1 x1 x2 Not upper hemicontinuous at x1 Not upper hemicontinuous at x2 Lower hemicontinuous Γ2 x1 x2 Not lower hemicontinuous at x1 Not lower hemicontinuous... (x)) Figure E.2 ε 2 y dH (A, B) B A dH (A, B) B ω(A, B) dH (A, B) A B ω(A, B) Figure E.3 A S co(S) S S is convex S co(S) S is not convex co(S) S S is not convex S is not convex Figure G.1 co(S) 0 0 0 Non-convex cones in R2 0 0 0 Convex cones in R2 0 0 0 Sets in R2 which are not cones Figure G.2 0 x y 0 C a base for C C 0 not a base for C Figure G.3 C 0 not a base for C y y x an algebraic interior point . C.2 f 3 f 4 f 6 1 1 1 6 1 4 1 3 1 2 Figure C.3 f x f(x) f 2 (x) x f(x) ¯x 45 ◦ line 45 ◦ line f f is a contraction with the unique fixed point ¯x. f is not a contraction, and it has no fi x ed points. Figure C.4 45 ◦ line f g. 1 f + ε f f m f − ε Figure D.5 f 1 f 2 f 3 f 4 1 1 Figure D.6 Figure D.7 1 2 7 8 1 f 1 f 2 f 3 -1 1 0 r(x) x Φ(x) B 2 Figure D.8 x 1 x 2 x 1 x 2 x 1 x 2 Γ 1 Γ 2 Γ 3 Not upper hemicontinuous at