Since, $T$ is Hausdorff, the graph of the continous linear map $i \circ f$ is closed subspace of $V \times T$. Again, the linear map, ${Id}_{V} \times i : V \times W \to V \times T$ is continuous and $({Id}_{V} \times i )^{1} (G_{i \circ f}) = G_f$ (since, $i$ is a injective map). \begin{align*}G_{f} \subseteq V \times W \overset{{Id}_{V} \times i}{\longrightarrow} G_{i \circ f} \underset{\text{(closed)}}{\subseteq} V \times T\end{align*} Hence, $G_f$ is a closed subspace of $V \times W$, i.e., by Closed Graph theorem it follows that $f$ is continuous.
