Understanding continuity

Dear Biswajit:



Thanks for the link. I thought a comment on continuity in a more abstract setting would be relevant here.



Let us consider a function f:V --> W, where V and W are two sets (for now with no structure defined on them). One can define continuity as soon as V and W are equipped with topologies (a topology on a set is the set of all open sets defined in the set. This may sound strange but it turns out one can define many different topologies on a given set. The standard topology on the real line is the one that takes open intervals as open sets. One can, for example, say any set is open and that would be the discrete topology, etc.) Having two topological sets, now f is continuous if and only if the preimage of any open set in W is open in V. One can show that the \epsilon-\delta definition is a special case of this.



Regards,

Arash