" 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.

For example,

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.

Thanks