How? To prove that is open, you need to take a point and show that there exists an such that . Have you done that?

After you understand why your proof is wrong, think of with .

What is open? You haven't specified any space, just the metric!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?