In the following reference (http://arxiv.org/abs/physics/0302045), the author tries to derive a general expression for coordinate transformation between inertial frames using just the assumptions of relativity, homogeneity of space and isotropy of space and time. However, in equation 12, he says the following $$\frac{\partial X}{\partial x} (x_1, t, v) = \frac{\partial X}{\partial x} (x_2, t, v)$$ and concludes that since the choice of $x_1$ and $x_2$ were arbitrary, the partial derivative is a constant. However, shouldn't he consider it dependent on $t$ and $v$ until proven otherwise?
I know that this is a similar question to this, but my question remains even after going through them. I think a better question to ask would be how homogeneity and isotropy imply linearity of coordinate transformations between inertial frames.
