  In this paper we consider the question of when the space $C_p(X)$ of
continuous real-valued functions on $X$ with the pointwise convergence topology
is countable dense homogeneous. In particular, we focus on the case when $X$ is
countable with a unique non-isolated point $\infty$. In this case, $C_p(X)$ is
countable dense homogeneous if and only if the filter of open neighborhoods of
$\infty$ is a non-meager $P$-filter.
