New inverse and implicit function theorems for differentiable maps with isolated critical points

New inverse and implicit function theorems for differentiable maps with isolated critical points

“…We first collect three lemmas. This result was established by the author [38]. We shall understand it from the viewpoint of maximal or minimal modulus.…”

“…Lemma 3.5 is the main result of the author's paper [38], which will appear soon in Real Analysis Exchange. Regarding the countable infinity assumption therein, the anonymous referee for the article suggested that it may be improved to zero one-dimensional Hausdorff measure.…”

“…Recall that H is a local homeomorphism on Ω\{x 0 }, so it follows from Theorem 2.2 that H is discrete at x 0 . Regarding the other question, it suffices to note the following open mapping theorem of the author [38]: Consequently, there exists an s > 0 and a (sufficiently small) r > 0 such that…”

Differentiable maps with isolated critical points are not necessarily open in infinite dimensional spaces

“…This result was rediscovered by the second-listed author [14] and is implied by Theorem 1 or 2 in [24] by Titus and Young. Furthermore, the second-listed author [15] showed that C 1 vector fields with isolated critical points are local homeomorphisms given that the dimension of the ambient space is higher than 2. For the interest of readers, we refer to [2,3,4,6,7,8,9,11,12,13,18,20,21,23] for some relevant works in Euclidean spaces.…”