I am moving my first steps in continuous model theory (tl;dr). This is one of two soft questions on the relation between a continuous and a classical structure.
Let $M$ be a classical first-order structure of signature $L$. For simplicity assume $M$ saturated an $L$ countable. Let $d$ be a pseudometric that induces the topology generated by the $0$-definable subsets of $M$. (It exists by Urysohn metrization theorem.)
Let $M^c$ be the continuous (pre)structure with the same support as $M$, the same functions as $M$, the pseudometric $d$ introduced above, and the canonical $\{0,1\}$-valued functions for the relations of $L$.
How is this related to the original first order theory? (I apologize for the open-endedness.)