It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. f\right|_{X_i}$ is continuous for all $i$ but $f$ is not continuous.
In Munkres book he places an additional condition in order to formulate an infinite pasting lemma for closed sets, that the collection $\{X_i\}$ be locally finite. What are other "nice" sufficient conditions that force the infinite closed pasting lemma to be true, or is this in some way "minimal"? To restrict the problem a little more we can let $X$ be a metric space since that's the context I'm working in.