Exercise 3, § 7

Let $X, X'$ be two topological spaces and $f : X \to X'$ a mapping which is continuous at a point $x_0 \in X$. Show that if $\mathfrak{B}$ is any filter base on $X$ which has $x_0$ as a cluster point, then $f(x_0)$ is a cluster point of the filter base $f(\mathfrak{B})$ on $X'$.