  The central purpose of this article is to establish new inverse and implicit
function theorems for differentiable maps with isolated critical points. One of
the key ingredients is a discovery of the fact that differentiable maps with
isolated critical points are discrete maps, which means that algebraic topology
methods could then be deployed to explore relevant questions. We also provide a
purely topological version of implicit function theorem for continuous maps
that still possesses unique existence and continuity. All new results of the
paper are optimal with respect to the choice of dimensions.
