Lusin and Suslin properties of function spaces

A topological space is $Suslin$ ($Lusin$) if it is a continuous (and bijective) image of a Polish space. For a Tychonoff space $X$ let $C_p(X)$, $C_k(X)$ and $C_{{\downarrow}F}(X)$ be the space of continuous real-valued functions on $X$, endowed with the topology of pointwise convergence, the compact-open topology, and the Fell hypograph topology, respectively. For a metrizable space $X$ we prove the equivalence of the following statements: (1) $X$ is $\sigma$-compact, (2) $C_p(X)$ is Suslin, (3) $C_k(X)$ is Suslin, (4) $C_{{\downarrow}F}(X)$ is Suslin, (5) $C_p(X)$ is Lusin, (6) $C_k(X)$ is Lusin, (7) $C_{{\downarrow}F}(X)$ is Lusin, (8) $C_p(X)$ is $F_\sigma$-Lusin, (9) $C_k(X)$ is $F_\sigma$-Lusin, (10) $C_{{\downarrow}F}(X)$ is $C_{\delta\sigma}$-Lusin. Also we construct an example of a sequential $\aleph_0$-space $X$ with a unique non-isolated point such that the function spaces $C_p(X)$, $C_k(X)$ and $C_{{\downarrow}F}(X)$ are not Suslin.