We extend Krivine's classical realizability to a simply typed calculus with some constants and primitives and a call-with-current-continuation construct similar to Felleisen's C operator. We show how the theory extends smoothly by associating an appropriate truth value to concrete types. As a consequence, results and methods from realizability still hold in this new context; this is especially interesting in the case of specification theorems because they hold a significant meaning from the programmer's point of view.