Gluing continuous functions defined on pieces

Suppose $F_{1}, \ldots , F_{n}$ are disjoint closed subsets of $\mathbb {R}$. Prove that if

is a function such that $\left.g\right|_{F_{k}}$ is a continuous function for each $k \in \left\{ 1, \ldots , n\right\}$, then $g$ is a continuous function.

Let $A$ and $B$ be measurable sets. Is $f: A \cup B \rightarrow \mathbb {R}$ continuous if and only if its restrictions to $A$ and $B$ are continuous?

Let $X$ and $Y$ be topological spaces and consider a mapping $f$ from $X$ to $Y$. Suppose $X = X_{1} \cup X_{2}$ and the restrictions of $f$ to the topological subspaces $X_{1}$ and to $X_{2}$ are continuous. Show that $f$ need not be continuous at any point in $X$. Show that $f$ is continuous on $X$ if $X_{1}$ and $X_{2}$ are open. Compare this with the case of measurable functions and the inheritance of measurability from the measurability of restrictions.