Why doesn't the Jacobian determinant $Jf$ change sign on $V_x$? Even if $Jf$ is continuous, it is of constant sign on a small neighborbood. How could the following calculation be legitimate? Why can we pull out the $\mathrm{sgn}$ from the inside of the integral? Thanks.
The function is a diffeomorphism on each such set $V_x$, and so the Jacobian is non-zero (hence of constant sign by connectedness).