Mapping from Discrete Space is Continuous

Theorem
Let $T_1 = \left({X_1, \vartheta_1}\right)$ be the discrete topological space on $X_1$.

Let $T_2 = \left({X_2, \vartheta_2}\right)$ be any other topological space.

Let $\phi: X_1 \to X_2$ be a mapping.

Then $\phi$ is continuous.

Proof
From the definition of continuous:
 * $U \in \vartheta_2 \implies \phi^{-1} \left({U}\right) \in \vartheta_1$

But as $\phi^{-1} \left({U}\right) \subseteq X_1$ it follows from the definition of discrete space that $\phi^{-1} \left({U}\right) \in \vartheta_1$.