Exercise 7.6.10

Note that since each $G_n$ is open, and the union of any collection of open sets is open, ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^N{G}_n$ is open, its complement is closed, and $K=[a,b]\cap {({\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^N{G}_n)}^c$ is closed. $K$ is also bounded so $K$ is compact.

By how we defined $G_n$, $f$ is $\alpha$-continuous pointwise on $K$. Taking it as fact that $\alpha$-continuity on a compact set implies uniform $\alpha$-continuity, since $K$ is compact, we have that $f$ is uniformly $\alpha$-continuous on $K$.