Are there more languages than functions? My gut says "no", but I don't know why.

For any function $f$ over strings on an alphabet, one can define a language in which every word is just the concatenation of a string $s$, a delimiter, and $f(s)$, or the empty string if $f(s)$ is undefined.  So there at least as many languages as functions.  How can we define a function for every language?  (Forgive my ignorance about languages, automata, etc.  I have not studied in that area.  Perhaps my question is poorly constrained.)
It depends on the signature of the function.  In computability theory we typically deal with functions with the signature $f:\{0,1\}^* \to \{0,1\}$.  There is a one-to-one correspondence between functions $f:\{0,1\}^* \to \{0,1\}$ and languages $L \subseteq \{0,1\}^*$: in particular, the function $f$ corresponds to the language $L=\{x : f(x)=1\}$, and the language $L$ corresponds to the function $f(x) = 1$ if $x \in L$, $f(x) = 0$ if $x \notin L$.