constant map is continuous

Let \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) be topological spaces and

\begin{align*} f: X \rightarrow& Y \\ x \mapsto& y \end{align*}

be a constant map. Then \(f\) is continuous

Suppose \(O \subseteq Y\) is open. then

\begin{align*} f^{-1}[O] = \begin{cases} \emptyset & \mbox{if } y \not\in O \\ X & \mbox{else } \\ \end{cases} \end{align*}

where both sets are open by definition of a topology.