Continuity_from_Union_of_Restrictions

Let $T_1$ and $T_2$ be topological spaces. Then $f: T_1 \to T_2$ is continuous if either $\displaystyle T_1 = \bigcup_{i \mathop = 1}^n V_i$ where each $V_i$ is closed in $T_1$, and the restriction $f \restriction_{V_i}$ is continuous for each $V_i$ or $\displaystyle T_1 = \bigcup_{i \mathop \in I} U_i$ where $I$ is any (possibly infinite) indexing set; each $U_i$ is open in $T_1$, and the restriction $f \restriction_{U_i}$ is continuous for each $U_i$.