Let X be a topological space. show that if there exists a continuous, non constant map from X to the integers with the discrete topology, then X is not connected.
So I know that connected subspaces of integers with the discrete topology are just points. Also the image of a connected space under a continuous map is connected.
Here is where my reasoining for the proof eludes me. If I take the inverse image of those points is it that I haven now created a separation in the inverse image thus showing that X is not connected? Or is it that since the image of a connected space under a continuous function is connected, but since this maps to a point, the function is therefore constant?