I have troubles with understanding a few things in a proof of implicit function theorem (Rosenlicht's Introduction to Analysis). It would be great if somebody could explain the highlighted parts to me. I know it's lengthy, sorry about it. I have problems with:
Green - Is there any intuitive way of explaining why we have chosen such numbers? Geometric interpretation maybe?
Red - I don't get it why $\frac{\partial f} {\partial y}$ is nonzero there.
Green: This is an artificial complication. Since $F_y$ is continuous and $F_y(a,b)=0$, the set $\{(x,y):|F_y(x,y)|<1/2\}$ is open and contains $(a,b)$. Thus there is also an open rectangular neighborhood $(a-h,a+h)×(b-k,b+k)$ that is contained in that set.
The complication comes from first taking a disk of radius $r$ contained in that set $\{|F_y|<1/2\}$ and then inscribing a rectangle inside that disk. There is no convincing reason to go first via circular neighborhoods.
Red: $F(x,y)=y-K·f(x,y)$, thus $F_y=1-K·f_y$ or $K·f_y=1-F_y$. Now since $|F_y|<1/2$, there is no way to get $f_y=0$, more, $|f_y|>1/(2K)$.