From Fourier transform to Laplace Transform
So it sounds like X(s) is the analytic continuation of X(f), given the proper restrictions, is that right?


that's what i would say.  be careful about expressing the function and its argument.
use a different "$X(\cdot)$" if you need to.  like: 
$$ X_\mathscr{L}(j 2 \pi f) = X_\mathscr{F}(f) $$
where 
$$ X_\mathscr{L}(s) \triangleq \mathscr{L}\Big\{ x(t) \Big\} = \int\limits_{-\infty}^{\infty} x(t) \, e^{-st} \, \mathrm{d}t $$ 
and 
$$ X_\mathscr{F}(f) \triangleq \mathscr{F}\Big\{ x(t) \Big\} = \int\limits_{-\infty}^{\infty} x(t) \, e^{-j2\pi f t} \, \mathrm{d}t $$