The full abstractness problem was raised by Robin Milner and the diculty of designing fully abstract semantics is well-known. A denotational semantics is said to be fully abstract if it is correct and complete. The correctness condition is usually easy to establish. However, it may be difficult to establish the completeness condition. In this paper we study the abstractness of denotational models designed with continuation semantics for concurrency (CSC). Due to the interplay between denotations and continuations, the full abstractness problem seems to be even more dicult in continuation semantics. Although several papers employ continuations in the denotational design of concurrent languages, we are not aware of any full abstractness result for a concurrent language designed with continuations. Therefore we introduce a new criterion that we name weak abstractness (which is specific of the CSC technique): it preserves the correctness condition, but relaxes the completeness condition of the classic full abstractness criterion. Weak abstractness may be useful for a wide class of denotational models designed with continuations in which full abstractness is difficult (or impossible) to achieve. To illustrate this approach we present a denotational model for a simple asynchronous concurrent language designed with continuations over metric spaces. We show that this denotational semantics is weakly abstract with respect to a corresponding operational model.