Exercise 19, § 5

Let $A$ be a dense subspace of a topological space $X$, and let $f$ be a continuous mapping of $A$ into a complete metric space $Y$. Show that the set of points of $X$ where $f$ has no limit (relative to $A$) is a meagre set in $X$ [consider, for each $n > 0$, the set of points $x \in E$ at which the oscillation $\omega(x; f)$ of $f$ is $< 1/n$].