The purpose of this work is to, introduce pseudo-convex functions and to describe some of their properties and applications. The class of all pseudo-convex functions over a convex set C includes the class of all differentiable convex functions on C and is included in the class of all differentiable quasi-convex functions on C. An interesting property of pseudo-convex functions is that a local condition, such as the vanishing of the gradient, is a global optimality condition. One of the main results of this work consists of showing that the Kuhn-Tucker differential conditions are sufficient for optimality when the objective function is pseudo-convex and the constraints are quasi-convex. Other results of this work are a strict converse duality theorem for mathematical programming and a stability criterion for ordinary differential equations.