Abstract:  We study the existence of critical points of a smooth function in a given domain. We are particularly interested in the case when the Conley index of the isolated invariant set is trivial for the gradient flow. The main result is that when the Conley index is trivial, if there exists a homeomorphism between the entry set and exit set with a certain boundary value property, then we can remove the critical points. And we also present some interesting corollaries. Furthermore, our results have an application to an area-preserving map.