View source for Continuous map/Claim: continuous iff continuous at every point

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==Statement==
A [[map]], {{M|f:X\rightarrow Y}} between two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} is ''continuous'' {{iff}} it is ''continuous at every point''.
Symbolically:
* {{MM|1=\left(\forall\mathcal{O}\in\mathcal{K}[f^{-1}(\mathcal{O})\in\mathcal{J}]\right)\iff\left(\forall x_0\in X\forall N\text{ neighbourhood to }f(x_0)[f^{-1}(N)\text{ is a neighbourhood of }x_0]\right)}}
==Proof==
</noinclude>[[File:IMG 20151122 220401.jpg|200px]]
* A quick proof I did on some scrap - click for full version
{{Requires writeup|See image}}<noinclude>
==Notes==
<references group="Note"/>
==References==
<references/>
{{Theorem Of|Metric Space|Topology}}
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