I am studying the first chapter of this book:
Topological Degree Theory and Applications
At page 13 of this document, Definition 1.2.5, it is essentially said that to define the degree of a continuous function $f$ we just approximate it by a function $g$ in $C^2$ and take its degree.
Now I can't see how to justify this from the previous proof:
we just showed that if $\Omega$ is open and bounded, $f\in C^2(\bar \Omega) $, $p \not \in f(\partial \Omega)$ and $|p'-p|<\operatorname{dist}(p,f(\partial \Omega)$ where $p',p $ are regular values of $f$ then
$$\operatorname{deg}(f,\Omega,p)=\operatorname{deg}(f,\Omega, p')$$
While what we actually need is that if $g_1,g_2 \in C^2(\bar \Omega)$ and $\max( {|g_1-f|,|g_2-f|)}< d(p, f(\partial \Omega))$ then
$$\operatorname{deg}(g_1,\Omega,p)=\operatorname{deg}(g_2,\Omega, p')$$
Essentially we need an homotopy invariance for smooth functions, or am I wrong?
Thanks!