It's true (after fixing the typo; as already noted, the $f:\Bbb C\to\Bbb R$ should be $f:\Bbb C\to\Bbb C$. Proving it is non-trivial, because first you need a definition of "inside $C$".
This is exactly why you're more likely to find language like "on and inside $C$" in elementary books like say Brown&Churchill; they're willing to just wave their hands. In more rigorous/"advanced" texts these days they tend to avoid the notion of the inside of a curve... (what needs to be proved after the definitions are in place is that if $V$ is the bounded open set bounded by $C$ then $C$ is in fact the topological boundary of $V$, or equivalently $\overline V = V\cup C$. Once you know that, the version of MMT you know gives the current version.)