Does this question even make sense?Prove that any infinte set with co-finite topology is connected.

Let be an infinite space with co-finite topology.

Let be a two point discrete space where .

Let be any continuous function.

I need to show that f is constant.

My big problem is this:

We know and are open.

Hence .

However, is finite so there exist finitely many points where or where .

By that reasoning, we have a finite number of points where the function is not defined. These points are in so the function is discontinuous at each .

Why is this right or wrong?