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Cloud infrastructures are evolving from centralised systems to geographically distributed federations of edge devices, fog nodes, and clouds - often known as the Cloud-Edge Continuum. Continuum systems are dynamic, often massive in scale, and feature disparate infrastructure providers and platforms; this greatly increase the complexity of developing and managing applications. The Serverless paradigm shows the potential to greatly simplify the process of building Continuum applications - however, current scheduling mechanisms for Serverless Continuum platforms pay little attention to reducing the energy consumption and improving the sustainability of function execution. This is a significant omission, made worse as computing nodes within a Continuum may be powered by renewable energy sources that are intermittent and unpredictable, making low-powered and bottleneck nodes unavailable.There is great opportunity to design a decentralized energy management scheme for scheduling Serverless functions that takes advantage of the different layers of the Continuum, such as IoT devices located at the Edge, on-premises clusters closer to the data sources, or directly on large Cloud infrastructures. To achieve this, we formally model a green energy-aware Serverless workload scheduling problem for the multi-provider Cloud-Edge Continuum. We then design stable matching based technique for decentralized energy management (utilising a distributed controller) that considers the availability of green energy nodes and the QoS requirements of Serverless functions. We prove the complexity, stability and termination of the proposed heuristic algorithm, and also compare its performance with baseline scheduling techniques.