Serverless computing has been establishing itself as a compelling paradigm for the development and of modern cloud-native applications. Serverless represents the next step in the evolution of cloud programming models, services and platforms, which is especially appealing due to its low management overhead, easy deployment, scale-to-zero and the promise of optimized costs. Recently, due to the advantages it offers, the serverless paradigm has been growing beyond traditional clouds, making its way to the Edge. The natural evolutionary step for serverless computing is to unify the Edge and the Cloud into what we refer to as Edge-Cloud Continuum. In this paper, we outline our vision of the Serverless Computing Fabric (SCF) for the Edge-Cloud continuum. We introduce the reference architecture for the SCF and show how it unlocks the full potential of the Edge-Cloud continuum. We also discuss main opportunities and challenges, which need to be overcome in order to achieve the vision of the Serverless Computing Fabric. Finally, we introduce key design principles together with core enabling runtime mechanisms, which are intended to serve as a research road map towards the Serverless Computing Fabric for Edge-Cloud continuum.