Difference between revisions of "Continuous map"

From Maths
Jump to: navigation, search
m
m (Continuous at a point)
Line 36: Line 36:
 
Claim: The [[mapping]] {{M|f}} is continuous {{M|\iff}} it is continuous at every point
 
Claim: The [[mapping]] {{M|f}} is continuous {{M|\iff}} it is continuous at every point
 
{{Begin Proof}}
 
{{Begin Proof}}
{{Todo|Do this}}
+
[[File:IMG 20151122 220401.jpg|200px]]
 +
* A quick proof I did on some scrap - click for full version
 +
{{Todo|Write up proof}}
 +
[[Category:Writeup]]
 
{{End Proof}}{{End Theorem}}
 
{{End Proof}}{{End Theorem}}
 +
 
==Sequentially continuous at a point==
 
==Sequentially continuous at a point==
 
Given two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}}, and a point {{M|x_0\in X}}, a [[function]] {{M|f:X\rightarrow Y}} is said to be ''continuous at {{M|x_0}}'' if<ref name="KMAPI"/>:
 
Given two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}}, and a point {{M|x_0\in X}}, a [[function]] {{M|f:X\rightarrow Y}} is said to be ''continuous at {{M|x_0}}'' if<ref name="KMAPI"/>:

Revision as of 22:33, 22 November 2015

This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.
Note: there are a few different conditions for continuity, there's also continuity at a point. This diagram is supposed to show how they relate to each other.

[math]\begin{xy}\xymatrix{ \text{Continuous} \ar@2{<->}[d]_-{\text{claim }1} \ar@2{<.>}[drr] & & \\ {\begin{array}{lr}\text{Continuous at }x_0\\ \text{(neighbourhood)}\end{array} } \ar@2{<->}[rr]_{\text{claim }2} & & {\begin{array}{lr}\text{Continuous at }x_0\\ \text{(sequential)}\end{array} } }\end{xy}[/math]

Note that:
  • All arrow denote logical implies, or "if and only if"
  • Dotted arrows show immediate results of the claims on this page
Overview Key

Definition

Given two topological spaces [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] we say that a map, [ilmath]f:X\rightarrow Y[/ilmath] is continuous if[1]:

  • [ilmath]\forall\mathcal{O}\in\mathcal{K}[f^{-1}(\mathcal{O})\in\mathcal{J}][/ilmath]

That is to say:

  • The pre-image of every set open in [ilmath]Y[/ilmath] under [ilmath]f[/ilmath] is open in [ilmath]X[/ilmath]

Continuous at a point

Again, given two topological spaces [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath], and a point [ilmath]x_0\in X[/ilmath], we say the map [ilmath]f:X\rightarrow Y[/ilmath] is continuous at [ilmath]x_0[/ilmath] if[1]:

UNPROVED: I suspect that this is the same as [ilmath]\forall\mathcal{O}\in\mathcal{K}[f(x_0)\in\mathcal{O}\implies f^{-1}(\mathcal{O})\in\mathcal{J}\wedge x_0\in f^{-1}(\mathcal{O})][/ilmath] - this is basically the same just on open sets instead




TODO: Investigate and prove the highlighted claim


(Leave any notes to self here)

Claim 1

Claim: The mapping [ilmath]f[/ilmath] is continuous [ilmath]\iff[/ilmath] it is continuous at every point


IMG 20151122 220401.jpg

  • A quick proof I did on some scrap - click for full version

TODO: Write up proof



Sequentially continuous at a point

Given two topological spaces [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath], and a point [ilmath]x_0\in X[/ilmath], a function [ilmath]f:X\rightarrow Y[/ilmath] is said to be continuous at [ilmath]x_0[/ilmath] if[1]:

  • [math]\forall (x_n)_{n=1}^\infty\left[\lim_{n\rightarrow\infty}(x_n)=x\implies\lim_{n\rightarrow\infty}(f(x_n))=f(x)\right][/math] (Recall that [ilmath](x_n)_{n=1}^\infty[/ilmath] denotes a sequence, see Limit (sequence) for information on limits)

Claim 2

Claim: [ilmath]f[/ilmath] is continuous at [ilmath]x_0[/ilmath] using the neighbourhood definition [ilmath]\iff[/ilmath] it is continuous at [ilmath]x_0[/ilmath] using the sequential definition




TODO: Fill this out


References

  1. 1.0 1.1 1.2 Krzysztof Maurin - Analysis - Part 1: Elements

Old page

First form

The first form:

[math]f:A\rightarrow B[/math] is continuous at [math]a[/math] if:
[math]\forall\epsilon>0\exists\delta>0:|x-a|<\delta\implies|f(x)-f(a)|<\epsilon[/math] (note the implicit [math]\forall x\in A[/math])

Second form

Armed with the knowledge of what a metric space is (the notion of distance), you can extend this to the more general:

[math]f:(A,d)\rightarrow(B,d')[/math] is continuous at [math]a[/math] if:
[math]\forall\epsilon>0\exists\delta>0:d(x,a)<\delta\implies d'(f(x),f(a))<\epsilon[/math]
[math]\forall\epsilon>0\exists\delta>0:x\in B_\delta(a)\implies f(x)\in B_\epsilon(f(a))[/math]

In both cases the implicit [math]\forall x[/math] is present. Basic type inference (the [math]B_\epsilon(f(a))[/math] is a ball about [math]f(a)\in B[/math] thus it is a ball in [math]B[/math] using the metric [math]d'[/math])

Third form

The most general form, continuity between topologies

[math]f:(A,\mathcal{J})\rightarrow(B,\mathcal{K})[/math] is continuous if
[math]\forall U\in\mathcal{K}\ f^{-1}(U)\in\mathcal{J}[/math] - that is the pre-image of all open sets in [math](A,\mathcal{J})[/math] is open.

Equivalence of definitions

Continuity definitions are equivalent