Difference between revisions of "Continuous map"
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==First form== | ==First form== | ||
The first form: | The first form: | ||
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==Equivalence of definitions== | ==Equivalence of definitions== | ||
[[Continuity definitions are equivalent]] | [[Continuity definitions are equivalent]] | ||
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+ | {{Definition|Topology|Metric Space}} |
Revision as of 10:47, 12 May 2015
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.