Let $X$ and $Y$ be topological spaces. Let $\vert Sing(-)\vert$ be the functor which sends a topological space to the (or "a"? there seem to be more possibylites, for me it's just important, that I get CW comlexes at the end) geometric realization of the singular set of the space.

Is the map $C(X,Y)\rightarrow C(\vert Sing(X)\vert,\vert Sing(Y)\vert)$ we get (as $\vert Sing(-)\vert$ is a functor) continuous in the compact-open topology?