If $\exists f : (X,\tau) \rightarrow (\{ 0,1\}, \tau_\text{discr})$ continuous, non constant, $(X,\tau) $ is not connected

Question

Let $(X, \tau)$ be a topological space and consider the discrete topology over $\{0,1\}$;

Prove that if, there exists a continuous non-constant mapping $ f : X \rightarrow \{ 0,1\}$, allora $(X, \tau)$ is not connected.

I have no idea how to tackle this, any idea?

Answer 1

Hint:

$X$ is connected if and only if the only both open and closed subspaces are $X$ and $\varnothing$. Try to find other open and closed subsets from these hypotheses.