Computation offloading is employed by mobile apps running over resource-constrained devices to leverage the cloud in overcoming their resource limits. The advent of the Multi-access Edge Computing (MEC) paradigm further extends the potential opportunities of mobile-cloud offloading, allowing new service provisioning scenarios, such as mobile gaming and multimedia, where responsiveness of mobile devices at the network edge significantly benefits from low latency interactions. However, state-of-the-art offloading platforms for MEC architectures have not addressed the technical challenge of supporting specific file systems for this MEC-enabled class of applications, with components running at three hosting environments, i.e., mobile, edge, and cloud. This paper proposes the Mobile Edge File System (MEFS), an application-level distributed file system designed to be highly resilient and able to efficiently maintain consistency among the mobile, edge, and cloud entities. MEFS supports application handoff through live migration as end devices move between edges. The cloud transparently helps with recovery from faulty edge nodes or in the case of unavailability of edges in the user's proximity. We implemented a MEFS prototype in Android along with MEFS-based MEC-enabled mobile apps. The experimental results show how MEFS can achieve low latency and low overhead.