Question: Let $f, g: U \rightarrow \mathbb{R}^n$ be differentiable functions in the open set $U\subset \mathbb{R}^m$, let $\delta$ be a real positive number and $X\subset U$. Defining '$|f-g|_{1} \leq \delta$ in $X$ ' meaning that $|f(x)-g(x)|\leq \delta$ and $|f'(x)-g'(x)| \leq \delta$ for every $x \in X$. Let $K\subset U$ be a compact subset and $f:U \rightarrow \mathbb{R}^n$ be a function of $C^1$ class, such that $f|K$ (restriction) is an immersion (that is, the derivative $f'(x): \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ is an injective function for every $x \in K$). Prove that exists $\delta>0$ such that if the function $g:U\rightarrow\mathbb{R}^{n}$ is $C^1$ class, with $|g-f|_{1}\leq \delta$ in $K$, then $g|K$ is an immersion.
My attempt: Let $A \subset \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)$ be an open subset made for every injective linear transformations from $\mathbb{R}^m$ to $\mathbb{R}^n$. We have $f':U \rightarrow \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)$ is continuous and $f'(K)\subset A$ is compact, where $A$ is open. Defining $\delta= distance\; [ f'(K), \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n) - A]>0$. We have $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)-A$ is a closed set. If $g:U\rightarrow \mathbb{R}^n$ and $|g-f|_{1}<\delta$, then $g'(K) \subset A$. Then $g|K$ is an immersion.
Remark: Please, would you help me to improve this?