I am trying to find out whether or not this is true.
Let be an arbitrary topological space, and let have the order topology. If are continuous, and if for a specific , then there exists an open set , containing , such that for all .
If I can show this, then my question in the other thread becomes easier.
So far, my intuition tells me that the best way to proceed would be to use the definition of continuity that says is continuous at a point if for each open set containing , we can find an open set , containing , such that . I haven't gotten much further than that, though.