This technical note addresses the synchronized region problem, which is converted to a more convenient matrix stability problem, for complex dynamical networks. For any natural number n, the existence of a network with n disconnected synchronized regions is theoretically proved and numerically demonstrated. This shows the intrinsic complexity of the network synchronization problem. Convexity characteristic of stability for relevant matrix pencils is further discussed. A smooth Chua's circuit network is finally discussed as an example for illustration.