We present an approach for radar-inertial odometry which uses a continuous-time framework to fuse measurements from multiple automotive radars and an inertial measurement unit (IMU). Adverse weather conditions do not have a significant impact on the operating performance of radar sensors unlike that of camera and LiDAR sensors. Radar's robustness in such conditions and the increasing prevalence of radars on passenger vehicles motivate us to look at the use of radar for ego-motion estimation. A continuous-time trajectory representation is applied not only as a framework to enable heterogeneous and asynchronous multi-sensor fusion, but also, to facilitate efficient optimization by being able to compute poses and their derivatives in closed-form and at any given time along the trajectory. We compare our continuous-time estimates to those from a discrete-time radar-inertial odometry approach and show that our continuous-time method outperforms the discrete-time method. To the best of our knowledge, this is the first time a continuous-time framework has been applied to radar-inertial odometry.