I am reading "Topological degree theory and applications" by O'Regan, Cho and Chen.
I am stuck on the start:
Consider $\Omega\subset \mathbb{R}^n$ open and bounded and let $f\in C^1(\bar \Omega)$, if $p \not \in f(\partial\Omega) $ and $J_f(p)\not = 0$ (det. of the Jacobian matrix at p) then for some $\epsilon_0>0$
$\displaystyle deg(f,\Omega,p) = \int _{\Omega} \phi_{\epsilon}(f(x)-p)J_f(x)dx$
for all $\epsilon \in (0,\epsilon_0)$
where we define
$\displaystyle \phi_{\epsilon}(x)=\displaystyle c\epsilon^{-n}e^{\displaystyle -\frac{1}{1-|\displaystyle \epsilon^{-1}x|^2}}$ if $|x|<1$ and is zero otherwise.
$c$ is just a constant such that the integral of $\phi_{\epsilon}$ is 1.
The first line of the proof of this fact says that if $f^{-1}(p)=\emptyset $ then it is obvious, but I can't really see why...
Can anyone help? Thank you!
Because if $f^{-1}(p)=\emptyset$ then there exists $\epsilon_0>0$ such that $|f(x)-p|>\epsilon_0$ for all $x \in \Omega$. Then for $\epsilon<\epsilon_0$ you have $\Phi_\epsilon(f(x)-p) = 0$ for all $x \in \Omega$ since $|f(x)-p|>\epsilon_0$ and $\Phi_\epsilon(y)=0$ for $|y|\ge \epsilon$.