Until recently, Internet of Things applications were mainly seen as a means to gather sensor data for further processing in the Cloud. Nowadays, with the advent of Edge and Fog Computing, digital services are dragged closer to the physical world, with data processing and storage tasks distributed across the whole Cloud-to-Thing continuum. Function-as-a-Service (FaaS) is gaining momentum as one of the promising programming models for such digital services. This work investigates the current research landscape of applying FaaS over the Cloud-to-Thing continuum. In particular, we investigate the support offered by existing FaaS platforms for the deployment, placement, orchestration, and execution of functions across the whole continuum using the Systematic Mapping Study methodology. We selected 33 primary studies and analyzed their data, bringing a broad view on the current research landscape in the area.