Deep neural networks (DNN) based video analytics have empowered many new applications, such as automated retail and smart city. Meanwhile, the proliferation of fog computing systems provides system developers with more design options to improve performance and save cost. To the best of our knowledge, this paper presents the first serverless system that takes full advantage of the client-fog-cloud synergy to better serve the DNN-based video analytics. Specifically, the system aims to achieve two goals: 1) Provide the optimal analytics results under the constraints of lower bandwidth usage and shorter roundtrip time (RTT) by judiciously managing the computational and bandwidth resources deployed in the client, fog, and cloud environment. 2) Free developers from tedious administration and operation tasks, including DNN deployment, cloud and fog's resource management. To this end, we design and implement a holistic cloud-fog system referred to as VPaaS (Video-Platformas-a-Service) to execute inference related tasks. The proposed system adopts serverless computing to enable developers to build a video analytics pipeline by simply programming a set of functions (e.g., encoding and decoding, and model inference). These functions are then orchestrated to process video streaming through carefully designed modules. To save bandwidth and reduce RTT at the same time, VPaaS provides a new video streaming protocol that only sends low-quality video to the cloud. The state-of-the-art hardware accelerators and high-performing DNNs deployed at the cloud can identify regions of video frames that need further processing at the fog ends. At the fog ends, misidentified labels in these regions can be corrected using a light-weight DNN model. To address the data drift issues, we incorporate limited human feedback into the system to verify the results and adopt incremental machine learning to improve our system continuously. The evaluation of our system with extensive experiments on standard video datasets demonstrates that VPaaS is superior to several state-of-the-art systems: it maintains high accuracy while reducing bandwidth usage by up to 21%, RTT by up to 62.5%, and cloud monetary cost by up to 50%. We plan to release VPaaS as open-source software to facilitate the research and development of video analytics.