Exercise 1, § 3

Let $X$ be a topological space and $Y$ a metric space. Show that the set of all mappings of $X$ into $Y$ whose oscillation (Chapter IX, § 2, no. 3) at every point of $X$ is $\leq \alpha$ (where $\alpha$ is a given real number $> 0$) is closed in the space $\mathscr{F}_u(X; Y)$.