Chapter 2.18, Problem 9E

Let $\left\{A_{\alpha }\right\}$ be a collection of subsets of $X$; let $X={\displaystyle {\cup }_{\alpha }{A}_{\alpha }}$. Let $f:X\to Y$; suppose that $f|A_{\alpha }$, is continuous for each $\alpha$.

$\left(\text{a}\right)$ Show that if the collection $\left\{A_{\alpha }\right\}$ is finite and each set $A_{\alpha }$ is closed, then $f$ is continuous.

$\left(\text{b}\right)$ Find an example where the collection $\left\{A_{\alpha }\right\}$ is countable and each $A_{\alpha }$ is closed, but $f$ is not continuous.

$\left(\text{c}\right)$ An indexed family of sets $\left\{A_{\alpha }\right\}$ is said to be locally finite if each point $x$ of $X$ has a neighborhood that intersects $A_{\alpha }$ for only finitely many values of $\alpha$. Show that if the family $\left\{A_{\alpha }\right\}$ is locally finite and each $A_{\alpha }$ is closed, then $f$ is continuous.