A random variable need not be continuous. A random variable is a measurable function
$$X:(\Omega,\mathscr{A})\to (\mathbb{R},\mathscr{B}).$$
A measurable function "transports" a measure from $(\Omega,\mathscr{A})$ to $(\mathbb{R},\mathscr{B})$ as follow
$$\mathbb{P}_X(B)=\mathbb{P}(X\in B),\quad A\in\mathscr{B}.$$
Now for a measurable function $u:\mathbb{R}\to\mathbb{R}$, you can integrate the composition $u\circ X$ with respect to this image measure as follows
$$\int_{\Omega} u(X(\omega))\mathbb{P}(d\omega)=\int_{\mathbb{R}}u(x)\mathbb{P}_X(dx).$$
Now a random variable is absolutely continuous, if its distribution, this image measure $\mathbb{P}_X$ is (absolutely) continuous with respect to the Lebesgue measure, meaning there exists a positive measurable, $\mathbb{P}_X$ almost everywhere uniqure $f$, such that:
$$\mathbb{P}_X(A)=\int_A f(x)\lambda(dx).$$
This $f$ is the density of $X$ or the Radon-Nikodym derivative of $\mathbb{P}_X$ with respect to $\lambda$.
Hence its characteristic function would be:
$$\phi_X(\xi):\mathbb{E}e^{i\xi X}=\int_\Omega e^{i\xi X(\omega)}\mathbb{P}(d\omega)=\int_\mathbb{R} e^{i\xi x}\mathbb{P}_X(dx)=\int_\mathbb{R}e^{i\xi x}f(x)\lambda(dx).$$