Difference between revisions of "Continuous map"
From Maths
(Created page with "{{Definition|Topology|Metric Space}} ==First form== The first form: <math>f:A\rightarrow B</math> is continuous at <math>a</math> if:<br /> <math>\forall\epsilon>0\exists\de...") |
(No difference)
|
Revision as of 00:28, 13 February 2015
First form
The first form:
f:A→B is continuous at a if:
∀ϵ>0∃δ>0:|x−a|<δ⟹|f(x)−f(a)|<ϵ (note the implicit ∀x∈A)
Second form
Armed with the knowledge of what a metric space is (the notion of distance), you can extend this to the more general:
f:(A,d)→(B,d′) is continuous at a if:
∀ϵ>0∃δ>0:d(x,a)<δ⟹d′(f(x),f(a))<ϵ
∀ϵ>0∃δ>0:x∈Bδ(a)⟹f(x)∈Bϵ(f(a))
In both cases the implicit ∀x is present. Basic type inference (the Bϵ(f(a)) is a ball about f(a)∈B thus it is a ball in B using the metric d′)
Third form
The most general form, continuity between topologies
f:(A,J)→(B,K) is continuous if
∀U∈K f−1(U)∈J - that is the pre-image of all open sets in (A,J) is open.
TODO: The most important Theorem, that these two continuity definitions are the same