It's the discrete topology. I have to proof that if I have $(X,\tau)$ and $(Y,\delta)$ two topological spaces, if every function $f:X\longrightarrow Y$ is continuos then $\tau$ is the discrete topology. I don't know what is the way to solve this. Thanks!

Let $U \subset Y$ be a nontrivial open subset of $Y$ (i.e., neither the empty set nor all of $Y$), with $u \in U$ and $v \in Y - U$, and let $S \subset X$ be any subset of $X$. Then continuity of the function $f: X \to Y$ defined by $$f(x) = \begin{cases} u & \text{ if } x \in S \\ v& \text{ if } x \notin S\end{cases}$$ witnesses the fact that $S$ is an open subset of $X$.
