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Let [itex]f:\bigcup_{\alpha}A_{\alpha} \rightarrow Y[/itex] be a function between the topological spaces Y and [itex]X=\bigcup_{\alpha}A_{\alpha}[/itex]. Suppose that [itex]f|A_{\alpha}[/itex] is a continuous function for every [itex]\alpha[/itex] and that [itex]{A_{\alpha}}[/itex] is locally finite collection. Suppose that [itex]A_{\alpha}[/itex] is closed for every [itex]\alpha[/itex].

Show that: [itex]f[/itex] is continuous.

Any hints?

I'm stuck with this problem for some days. Some gave me answers on mathematics stackexchange. but it didn't make much sense.