Pasting Lemma/Counterexample of Infinite Union of Closed Sets

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

Let $I$ be an Definition:Infinite index set.

For all $i \in I$, let $C_i$ be closed in $T$.

Let $f: X \to Y$ be a mapping such that the restriction $f \restriction_{C_i}$ is continuous for all $i$.

Then:
 * $f$ need not be continuous on $C = \ds \bigcup_{i \mathop \in I}C_i$, that is, $f \restriction_C$ need not be continuous.