Say $f$ is a piecewise smooth function that is $f(x,y;\chi)=f_{1}(x,y;\chi)$ if $y\leq g(x;\chi)$ and $f(x,y;\chi)=f_{2}(x,y;\chi)$ if $y \geq g(x;\chi)$
where $g$ is a $C^2$ function and $\chi$ a parameter.
Now I am trying to understand this statement -
If the product of $\det(Df_{1})_{|(0,1;0)}$ and $\det(Df_{2})_{|(0,1;0)}$ is positive, then $f$ is locally invertible
Well if the product of the determinant is positive then neither of the determinants is non-zero and hence locally invertible by Inverse function theorem
But I am thinking whether saying this would be wrong?
If the product of $det(Df_{1})_{|(0,1;0)}$ and $det(Df_{2})_{|(0,1;0)}$ is negative or non-zero, then $f$ is locally invertible?
Also Can we generalize this?