Continuous map/Claim: continuous iff continuous at every point

Statement

A map, [ilmath]f:X\rightarrow Y[/ilmath] between two topological spaces [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] is continuous if and only if it is continuous at every point. Symbolically:

• [math]\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)[/math]

Proof

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

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This requires one or more proofs to be written up neatly and is on a to-do list for having them written up. This does not mean the results cannot be trusted, it means the proof has been completed, just not written up here yet. It may be in a notebook, some notes about reproducing it may be left in its place, perhaps a picture of it, so forth. The message provided is:
See image

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