The effect of calmness on the solution set of systems of nonlinear equations

We address the problem of solving a continuously differentiable nonlinear system of equations under the condition of calmness. This property, also called upper Lipschitz-continuity in the literature, can be described by a local error bound and is being widely used as a regularity condition in optimization. Indeed, it is known to be significantly weaker than classic regularity assumptions that imply that solutions are isolated. We prove that under this condition, the rank of the Jacobian of the function that defines the system of equations must be locally constant on the solution set. In addition, we prove that locally, the solution set must be a differentiable manifold. Our results are illustrated by examples and discussed in terms of their theoretical relevance and algorithmic implications.