Pasting Lemma/Continuous Mappings on Open Sets

Theorem
Let $T = \struct {X, \tau}$ and $S = \struct {Y, \sigma}$ be topological spaces.

Let $I$ be an indexing set.

Let $\family {C_i}_{i \mathop \in I}$ be a family of open sets of $T$.

Let $\family {f_i : C_i \to Y}_{i \mathop \in I}$ be a family of continuous mappings.

Let $C = \ds \bigcup_{i \mathop \in I} C_i$.

Let $f = \ds \bigcup_{i \mathop \in I} f_i : C \to Y$ where $\ds \bigcup_{i \mathop \in I} f_i$ is the union of relations.

Let for all $i, j \in I$, $f_i$ and $f_j$ agree on $C_i \cap C_j$.

Then $f$ is a continuous mapping on $C = \ds \bigcup_{i \mathop \in I} C_i$.

Proof
From Union of Family of Mappings which Agree is Mapping:
 * $f$ is a mapping from $C$ to $Y$.

From Restriction of Union of Mappings which Agree Equals Mapping:
 * $\forall i \in I : f \restriction_{C_i} = f_i$

Hence:
 * $\forall i \in I : f \restriction_{C_i}$ is continuous

From Pasting Lemma for Union of Open Sets
 * $f$ is continuous

Also see

 * Pasting Lemma for Continuous Mappings on Closed Sets for an analogous statement for closed sets.