Summary: Every local minimizer of a smooth constrained optimization problem satisfies the sequential approximate Karush-Kuhn-Tucker (AKKT) condition. This optimality condition is used to define the stopping criteria of many practical nonlinear programming algorithms. It is natural to ask for conditions on the constraints under which AKKT implies KKT. These conditions will be called strict constraint qualifications (SCQs). In this paper we define a cone-continuity property (CCP) that will be shown to be the weakest possible SCQ. Its relation to other constraint qualifications will also be clarified. In particular, it will be proved that CCP is strictly weaker than the constant positive generator constraint qualification.