I am trying to solve the following exercise in a book:
Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded subset with smooth boundary, and let $f:\bar \Omega \to \mathbb{R}^n$ be $C^1$. Suppose that $p \notin f(\partial \Omega)$ is a regular value for $f$.
Then there exists an $\epsilon > 0$ such that for all $g\in C^1(\bar \Omega)$, with $\|f-g\|_{C^1}<\epsilon$, $p$ is a regular value for $g$.
How can I prove this?
Naively, all I can see is that if $g(x)=p$, then we have $|f(x)-p|<\epsilon, |df_x-dg_x|<\epsilon$. I tried to prove that $x$ must lie in a small neighbourhood of a point in $f^{-1}(p)$, but I don't see how to do that. I am not even sure if this is true.
Any help or hints would be appreciated.
Consider the compact (submanifold with boundary) $S=f^{-1}(p)\subset \bar{\Omega}$. I will break the proof in steps and will leave you the detail:
There exists $\delta>0$ and an open neighborhood $U$ of $S$ in $\bar{\Omega}$ such that $||\nabla f(x)||\ge \delta$ for all $x\in U$.
There exists $\eta>0$ such that $||f-g||_{C^0}<\eta$, $g\in C^0(\bar \Omega)$ implies that
$$ g^{-1}(p)\subset U. $$
For every $g\in C^1(\bar \Omega)$, $||f-g||_{C^1}<\delta/2$ implies that $||\nabla g- \nabla f||_{C^0(U)}<\delta/2$.
Conclude the proof by working out a value of your $\epsilon$ in terms of $\delta$ and $\eta$.