Let $f,g:V \rightarrow \mathbb{R}^n$ differentiable in the open $ V \subseteq \mathbb{R}^m $ let $ K \subseteq V $compact and $ f: V \rightarrow \mathbb{R}^n $ of class $ \mathcal{C}^1 $ such that $ f|_K $ is a immersion ($ f'x: \mathcal{R}^m \rightarrow \mathbb{R}^n $ is injective for all $x$ in $K$) show that there exists a $\delta> 0 $ such that if application $ g:V \rightarrow \mathbb{R}^n $ of class $ \mathcal{C}^1 $ with $ || g-f ||_1 <\delta $ then $ g|_K $ is a immersion.
$ || g-f || _1 <\delta $ is equivalent to $ | g-f | <\delta $ and $ | g'-f '| <\delta$, for this existence, I define $ h = g-f $ with $ h '= g'-f' $ since $ g, f $ is of class $ \mathcal{C}^1 $ then $ h $ is of class $ \mathcal{C}^1 $ as $ K $ is compact of $ \mathbb{R}^{n} $ so we have that $ h $ and $ h'$ are bounded, therefore there exists $ \delta_1, \ \delta_2> 0 $ such that $ | g-f | <\delta_1 $ and $ | g'-f '| < \delta_2 $ if we take $ \delta = \max\{\delta_1, \delta_2\}$ we have $ | g-f | <\delta $ and $ | g'-f' | <\ delta $ therefore $|| g-f ||_1 <\delta $, for the implication of $ g $ is a immersion I thought to prove that the $Ker(g') = {0}$, showing that there exists $ M> 0 $ such that $| Dg_x (h) |>M$ for all $ h \in K - \{0 \}. $ but I can't advance. Thanks