The notion of Continuity is one of the most important in real analysis, partly because continous functions most closely resemble the behaviour of observables in nature.
Although continuity and continous functions can be defined on more general sets, we will first restrict ourselves to
We say that is continous at point if such that for all ,
If is continous at for all , we say that is continous over .
Definition for metric spaces
We can easily extend this definition to metric spaces. Let and be metric spaces. Given a function , and a point , we say that is continuous a if, for all there is a such that for all ,
If is continous at for all , we say that is continous over
Definition for Topological spaces
Perhaps the most general definition of continuity is in the context of topological spaces. If and are topological spaces, then a function is called continuous if for any open set in , it's preimage (i.e. the set ) is an open set in . Note that the image of an open set in does not have to be open.
It can be shown that if and are metric spaces under the metric space topology, that this definition of continuity coincides with the previous one.
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