I think that a). is valid. The definition of continuity, at a point , adapted to this case is:Is at least one of the following two implications valid for every map of a metric space to a metric space ?
a). is continuous is open for every .
b). is open for every open is continuous.
is just valid .
Hence it does imply that is open for every .
So, i've answered the question with a yes! However, for the sake of curiosity I will try and figure out b).
Since a). is true, i'll try and find a counterexample.
Consider the French Railway Metric. This is open because a ball can be constructed around every point.
However, is not continuous.
Suppose that and .
As , so it cannot be continuous.
Is this the correct way to think about the problem?