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Let $X = A \cup B$, where $A$ and $B$ are subspaces of $X$. let $f: X \to Y$; suppose that the restricted functions $f\upharpoonright A:A\to Y$ and $f\upharpoonright B:B\to Y$ are continuous. Show that if both $A$ and $B$ are closed in $X$, then $f$ is continuous.

How does using h and g, as arbitrary functions in the hint below work?

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    What is $X$? A metric space? A subset of $\mathbb{R}^n$?2011-11-06
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    If $X$ is a general topological space, $\epsilon-\delta$ arguments aren’t available: they require a metric. (They’re also unnecessarily complicated, unless the only definition of continuity that you have is for metric spaces.)2011-11-06
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    Use the fact that continuity can be equivalently defined as: the inverse image of a closed set is closed.2011-11-06

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