The aim of this paper is to extend the modal logic of knowledge due to Moss and Parikh by state transformers arising, eg, from actions of agents. The peculiarity of Moss and Parikh's approach lies in the fact that topological reasoning is supported. Emphasizing this we are mainly concerned with the idea of continuity here, which can be captured with the aid of the extended framework. We define and discuss an appropriate language, and we study the accompanying logical system. It turns out that certain expressive means from a particular extension of the commonly employed modal formalism, viz hybrid logic, are very useful for our purposes. With that, we obtain the finite axiomatizability, soundness and completeness, and decidability, of the new logic.