Continuous logic is a relatively new logic better equipped for studying the model theory of structures based on complete metric spaces. There are continuous analogs of virtually every notion and theorem from classical model theory, often with equalities replaced by approximations. However, most of the work done in continuous logic has centered around sophisticated topics concerning stability and its generalizations. In this talk, I will discuss the more basic notion of definability in metric structures. More specifically, I will consider the question of which functions are definable in Urysohn's metric space. Urysohn's metric space is the unique (up to isometry) Polish space that is universal and ultrahomogeneous. In many ways, Urysohn's metric space is to continuous logic as the the infinite set is to classical logic. However, we will see that the task of understanding the definable functions in Urysohn's metric space involves some interesting topological considerations.