An interesting discussion about the distinction between callbacks and continuations over on SO has prompted this question. By definition, a continuation is an abstract representation of the logic needed to complete a computation. In most languages this manifests as a one argument procedure to which you pass whatever value needs continued processing.
In a purely functional language (where all functions are pure and first class citizens), I would think a continuation could be entirely modeled as a function. This is, after all, how I've previously understood continuations up to this point. However, the world is full of state (sigh..) and so the general definition does not require that a continuation capture program state -- it need only encompass intent.
To help my understanding, can an example be provided in a functional language where the continuation is expressed in a more abstract way than a function? I know Scheme allows you to grab the current continuation in a first-class manner (call/cc), but even so, it seems that the one argument procedure passed to call/cc is simply given the current continuation in the form of another one argument procedure to which the call/cc'd function can apply its result.