" is continuous if and only if is open whenever is an open set."
When is it best to use this definition and what would the steps be to use it?
I was thinking the implication "if is open whenever is an open set, then is continuous," might be helpful for counter examples.
suppose and (floor function).
Do I have produceds an open set (where is a -neighborhood centered at some limit point c of the domain of )?
Could I use it to show is not coninuous at zero?
I can't find many examples of this characterization of countinuity, so I am finding it hard to work with. I would like to know how to use it for practical continuity problems, as in, showing a particular function is continuous on some interval.
Notice an important distinction between this definition of "continuous" and the one from Calculus I: in Calculus I we define "continuous at a point" and then say that a function is continuous "on a set" if and only if it is continous at each point of that set. Here, we are defining "continuous" on the entire space. There is no way to talk about a function that is "continous" at some points in the space and not at others.