The optimal filtering problem for systems subject to both the discrete and continuous measurements is studied. The observability and detectability properties of such systems are investigated pointing out their connections with the existence of stable discrete-continuous observers. The optimal filter is based on the solution of a suitable matrix differential Riccati equation with jumps. Sufficient and necessary conditions for the existence of periodic stabilizing solutions to such an equation are worked out. The main result states that both detectability and stabilizability are necessary and sufficient for the existence of a unique periodic solution which is also stabilizing. Stabilizability and detectability also guarantee asymptotic convergence of the Kalman filter to the steady-state periodic filter irrespective of the initial state covariance. The results are illustrated by means of a numerical example.