Pseudocompact spaces $X$ and $df$-spaces $C_{c}\left( X\right)$

Let $X$ be a completely regular Hausdorff space, and let $C_{c}\left( X\right)$be the space $C\left( X\right)$ of continuous real-valued functions on $X$endowed with the compact-open topology. We find various equivalent conditions for $C_{c}\left( X\right)$ to be a $df$-space, resolving an old question of Jarchow and consolidating work by Jarchow, Mazon, McCoy and Todd. Included are analytic characterizations of pseudocompactness and an example that shows that, for $C_{c}\left( X\right)$, Grothendieck's $DF$-spaces do not coincide with Jarchow's $df$-spaces. Any such example necessarily answers a thirty-year-old question on weak barrelledness properties for $C_{c}\left( X\right)$, our original motivation.