We explore aspects of continuity as they manifest in two separate settings - metric model theory (continuous logic) and enriched categories - and interpret the former into the latter. One application of continuous logic is in proving that certain convergence results in analysis are in fact uniform across the choices of parameters: Avigad and Iovino outline a general method to deduce from a given convergence theorem that the convergence is uniform in a metastable'' sense. While convenient, this method imposes strict requirements on the kinds of theorems allowed: in particular, any functions occurring in the theorem must be uniformly continuous. In aiming to apply to a broader class of examples the Avigad-Iovino approach, we construct a variant of continuous logic that is able to handle discontinuous functions in its domain of discourse. This logic weakens the usual continuity requirements for functions, but compensates by introducing a notion of linear structure'' that mimics e.g. the vector space structure of Banach spaces. We use this logic to apply the Avigad-Iovino method to specific convergence results from functional analysis involving discontinuous functions, and obtain uniform metastable convergence in those examples. This is the project of the first part of this thesis. The second part of the thesis continues this study of continuity from a different angle, starting from where Lawvere shows that enriching a category over R with the appropriate monoidal structure turns that category into a metric space. He even muses on the notion of an R-valued logic'', but does not make the connection to continuous logic (primarily because continuous logic did not yet exist). We introduce necessary structure that enables us to have a notion of uniform continuity'' and ``continuous subobjects'' in an enriched categorical setting, and use this to give an interpretation of continuous logic into a certain category of R-enriched categories.