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We propose a novel strategy for energy-efficient dynamic computation offloading, in the context of edge-computing-aided beyond 5G networks. The goal is to minimize the energy consumption of the overall system, comprising multiple User Equipment (UE), an access point (AP), and an edge server (ES), under constraints on the end-to-end service delay and the packet error rate performance over the wireless interface. To reduce the energy consumption, we exploit low-power sleep operation modes for the users, the AP and the ES, shifting the edge computing paradigm from an always on to an always available architecture, capable of guaranteeing an on-demand target service quality with the minimum energy consumption. To this aim, we propose an online algorithm for dynamic and optimal orchestration of radio and computational resources called Discontinuous Computation Offloading (DisCO). In such a framework, end-to-end delay constraints translate into constraints on overall queueing delays, including both the communication and the computation phases of the offloading service. DisCO hinges on Lyapunov stochastic optimization, does not require any prior knowledge on the statistics of the offloading traffic or the radio channels, and satisfies the long-term performance constraints imposed by the users. Several numerical results illustrate the advantages of the proposed method.