All the references that I could find on the proof of the linearity of the transformations assume that the map that connects the coordinates between two inertial frames is at least once differentiable. I was curious if there was a way to get rid of the differentiability assumption.
For example, it can be shown that any map that preserves a non-degenerate bilinear form is bound to be linear. So the transformations, that must preserve the Minkowski product, must be linear. This reasoning, however, seems to be circular since all the proofs that I have seen on the invariance of the Minkowski product assume linearity! However this reasoning shows that one could remove the differentiability hypothesis by showing that the product is preserved, but without assuming linearity.
Thanks in advance for any help.