Mainstream Mathematics Proximity & Continuity Jeroen Goudsmit - PowerPoint PPT Presentation

Mainstream Mathematics Proximity & Continuity Jeroen Goudsmit Utrecht University friday november th, 

. . . q p

. . . . . q − p q 0 p

. . . . q − p d ( p, q ) = ∥ p − q ∥ q 0 . p

Open Ball . . . p

Open Ball . . . q p .

Open Ball . . . q d ( p, q ) p .

Open for all there is an such that . U ⊆ R n is open when

Open U ⊆ R n is open when for all p ∈ U there is an ϵ > 0 such that B ( p ; ϵ ) ⊆ U .

Limit Point each neighbourhood of intersects p is a limit point of A when

Limit Point p is a limit point of A when each neighbourhood of p intersects A

Proximity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 100 98 99 82 81 80 78 77 83 93 97 84 92 86 87 96 88 95 94 76 89 90 91 85 79 70 72 73 75 74 71 65 69 68 66 63 64 61 55 60 59 56 58 62 67 57 47 51 46 48 49 54 50 45 53 52 38 39 40 43 44 41 42 34 37 36 33 35 29 30 32 31 27 28 26 25 24 22 23 21 20 19 17 18 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Limit . . . . . R n f A

Limit . . . . . . . R n f A p q x → p f ( x ) = q lim

Limit . . . . . . . . R n f A p q B ( q ; ϵ ) x → p f ( x ) = q lim

Limit . . . . . . . . R n f A p q B ( p ; δ ) B ( q ; ϵ ) x → p f ( x ) = q lim

Continuity . . . . . continuous at R n f A

Continuity . . . . . . R n f A p f continuous at p

Continuity . . . . . . . . R n f A p f ( p ) B ( f ( p ) ; ϵ ) f continuous at p

Continuity . . . . . . . . R n f A p f ( p ) B ( p ; δ ) B ( f ( p ) ; ϵ ) f continuous at p