Bound constrained global optimization problems can be solved by interval branch-and-contract algorithms. These algorithms mainly combine branching steps and interval contractors to process the initial box defined by the bound constraints. These contractors apply constraint propagation techniques or interval operators to constraints that must be verified by the global minimizers, hence eliminating non optimal parts of boxes. We introduce here an interval contractor specifically designed to handle the boundary of the initial box where a minimizer may not be a stationary point. This contractor exploits the optimality conditions and subsumes the classical monotonicity test based on interval arithmetic. It is implemented in an interval branch-and-contract algorithm and the experimental results on a set of standard benchmarks show substantial improvements.