Abstract

It is shown that – assuming a continuous function \(f: \mathbb{R}^n\to \mathbb{R}^m\) with an upper semicontinuous approximate Jacobian mapping \(Jf\) and invertible matrices \(A\in \overline{\text{conv }} Jf(x_0)\cup\text{conv}(J^\infty f(x_0)\setminus\{0\})\) – there exist numbers \(\delta> 0\) and \(\varepsilon> 0\) such that \[ \| f(x_0+ h)- f(x_0)\|\geq \varepsilon\| h\|\quad\text{for all }h\neq 0, \| h\|< \delta \] and \[ f(x_0)+ {\varepsilon\delta\over 2}\text{ int } B(0,1)\subseteq f(x_0+ \delta\text{ int }B(0, 1)). \] As corollaries of this result the authors derive associated inverse and implicit function theorems.