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2021 Open and surjective mapping theorems for differentiable maps with critical points
Liangpan Li
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Real Anal. Exchange 46(1): 107-120 (2021). DOI: 10.14321/realanalexch.46.1.0107

Abstract

Let $\Omega$ be an open subset of $\mathbb R^n (n \ge 2)$, and let $F : \Omega \rightarrow \mathbb R^n$ be a continuously differentiable map with countably many critical points. We show that $F$ is an open map. Let $G :\mathbb R^n \rightarrow \mathbb R^n (n \ge 1)$ be a continuously differentiable map such that $G(x) \rightarrow \infty$ as $x \rightarrow \infty$. Then it is proved that $G$ is surjective if and only if each connected component of the complement of the set of critical values of $G$ contains at least one image of $G$. Several applications of both theorems especially to complex analysis are presented.

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Liangpan Li. "Open and surjective mapping theorems for differentiable maps with critical points." Real Anal. Exchange 46 (1) 107 - 120, 2021. https://doi.org/10.14321/realanalexch.46.1.0107

Information

Published: 2021
First available in Project Euclid: 14 October 2021

Digital Object Identifier: 10.14321/realanalexch.46.1.0107

Subjects:
Primary: 26B99
Secondary: 30G30

Keywords: critical point , differentiable map , open map , polyanalytic function , surjective map

Rights: Copyright © 2021 Michigan State University Press

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Vol.46 • No. 1 • 2021
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