Multi-access edge computing achieves better application responsiveness by offloading tasks from end devices to edge servers installed at the vicinity. Practical scenarios, such as post-disaster rescuing and battlefield monitoring, make it attractive to use end devices themselves as edge servers. This, however, introduces a new challenge: Due to mobility and power limitation, the set of edge servers becomes dynamic. As some servers fail, the tasks that run on them will also fail. This paper introduces a new dynamic edge computing model and conducts the first study on robust task offloading which is tolerant to $h$ server failures. We propose online primal-dual algorithms that offload tasks as they arrive. We evaluate the performance of our robust task offloading solutions through extensive simulations based on real task sets. The results show that our proposed solutions can well handle edge dynamics and achieve near optimal throughput (above 95 percent) compared to the optimal offline benchmark algorithm.