  We prove that the locally convex space $C_{p}(X)$ of continuous real-valued
functions on a Tychonoff space $X$ equipped with the topology of pointwise
convergence is distinguished if and only if $X$ is a $\Delta$-space in the
sense of \cite {Knight}. As an application of this characterization theorem we
obtain the following results: 1) If $X$ is a \v{C}ech-complete (in particular,
compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered. 2)
For every separable compact space of the Isbell--Mr\'owka type $X$, the space
$C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals
$[0,\omega_1]$, then $C_p(X)$ is not distinguished. We observe that the
existence of an uncountable separable metrizable space $X$ such that $C_p(X)$
is distinguished, is independent of ZFC. We explore also the question to which
extent the class of $\Delta$-spaces is invariant under basic topological
operations.
