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?