Let \(X\) and \(Y\) be topological spaces and \(f\colon X\to Y\) continuous . If a subspace \(T\subseteq Y\) contains \(f(X)\) the function \(f\colon X\to T\) is continuous with respect to the subspace topology.

Proof
Apply the characteristic property to the subspace \(T\) of \(Y\).