This paper explores the broad area of Smart City services and how the evolving Cloud-Edge-IoT continuum can support application deployment in Smart Cities. We initially introduce a range of Smart City services and highlight their computational needs. We then discuss the role of the Cloud-Edge-IoT continuum as a technological platform to meet those needs. To validate this approach, we present the COGNIFOG platform, a Cloud-Edge-IoT platform developed to support city-centric use cases, and an initial technology trial that shows the early benefits of using the platform. We conclude with plans for improvements to COGNIFOG based on the trials and with a broader set of observations on the future of the Cloud-Edge-IoT continuum in Smart City services and applications.