In first-order logic, the notion of stability has been a driving force of Model Theory in the last decades since Shelah introduced it. A most relevant connection occurs in first-order logic: stability and definability are equivalent. The classical definition of stability involves the computation of cardinalities of spaces of types. However, there are several equivalent definitions, most notably "no formula has the order property". We will present another approach to stability using double limit conditions which is more suitable for continuous logics. Using results from C${}_{p}$-theory, , i.e. the topology of real-valued function spaces, we will show connections among double limit conditions, stability and definability in various continuous logics. As an application, we will expand some work of Casazza and Iovino concerning Gower's problem on the definability of pathological Banach spaces not including isomorphic copies of $l^p$ or $c_0$ in compact logics to stablish similar undefinability results for (continuous) ${\mathcal{L}}_{{\omega }_1,\omega }$. We will also discuss further lines on research in this direction.