Who cares about continuity?

From the paradox of apparent motion to the traditions of “continuity editing” in the classical age, to the later narratological uses of “continuity” and “continuity error”, film discourse leverages the analytical notions of continua and continuous maps as a stopgap between form and phenomenology. For all the legwork continuity does in the media discourse, there is little engagement with its analytical or topological dimensions (the actual meaning of the word). Reading film-theoretic continuity discourse with an eye to how it is informed by (and departs from) its formulation in math and science, it is not so much that film theorists have “misused” the word; rather, I contend, this proliferation of “continuities” speaks to a history of filmmakers and theorists striving to inculcate motion-picture editing with an aesthetics of the analytical.

 It is for this reason that I hope to write a critical history of “continuity” in film culture which tracks how the word acquired its multipedal cinematic meanings with an attention to its contemporaneous use in science and industry. In the most extreme, the use of “continuity” in discourses of film form reduces to an incantation which ascribes intelligence and modernity to particular standards of composition, and, by proxy, to those people “intelligent and modern” enough to have an appropriately sophisticated phenomenological experience upon viewing. Searching for mathematical consistency in how the media-discourse deploys “continuity” reduces the term to an analytical “appeal to the nature of motion” fallacy. “Continuity”, which we all claim to understand intuitively, is as much an inherent property of the moving-image as “civilization” or “virtue” is for the written word.

So what is continuity actually?

A function f is defined as continuous at a point x if for every δ>0 there exists some ε>0 for which for all y satisfying 0<|x-y|<ε, it holds that |f(x)-f(y)|<δ.

This is the definition people learn in their first Calculus class, but it is actually a specific case of a broader notion of continuity here applied to metric spaces (that is, spaces wherein it makes sense to talk about the difference between two points/numbers). When we work with any topological space, we can use a much more elegant definition of continuity. Continuous maps preserve topology.

What is a topology?

A topology is a set of rules on a space by which we decide which subsets of the space are called “open”. So, we say f is continuous if the pre-image of open sets in the codomain is always open in the domain.

Why should we care about topology when we talk about film editing?

People love talking about editing in terms of “continuity”. They will claim that certain series of shots “preserve continuity” while others “disturb” or “break” continuity. I contend that, in most cases, film theorists and editors are actually talking about certain topologies on spaces of still frames and spaces of short takes. There is no universal set of rules for what “continuity” means in film theory, and looking at continuity analytically reveals how the discourse works to disguise industry standards of composition as intuitive rules of editing born of phenomenology.