Einstein's Theory of Special Relativity - a mathematical mistake!

Einstein writes in \textit{The special and general relativity} (2006 Swedish edition, p.125, Appendix):"A light signal that runs along the positive x-axis propagated according to the equation x=ct". In mathematics the positive x-axis is defined as X_+ = {(x,0,0) | x >= 0}, i.e. a ray from the origin, not the whole line -inf < x < inf. Therefore the physical light signal exists only for x >= 0, x' >= 0 and t >= 0, t' >= 0. With the word "generally" Einstein extrapolates relation (5) x'-ct' = lambda(x-ct) to hold for all x, -inf < x < inf and all x', -inf < x' < inf, including t<0, t'<0 and x<0, x'<0 where no light signal exists for the case described in (3). This is not a logical consequence. Infinitely many functions satisfy the condition, e.g. (x'-ct')=(x-ct)^3. Two cases are distinguished: Case A: Keep physical domains. Then (5) is valid for x >= 0, x' >= 0 and (6) for x <= 0, x' <= 0. Intersection is only x=x'=0. Adding (5) and (6) gives only x=x'=0, the trivial case where both frames coincide. One cannot speak of two frames moving with v>0. Case B: Accept extrapolation via "generally". Then both (5) and (6) hold for -inf < x < inf and -inf < x' < inf. Addition is mathematically allowed, but it is performed in a new postulated system, not derived from the original light signals. Physical justification for extending a concrete signal emitted at t=0, x=0 to t<0, t'<0 and to all x, x' is missing in [Einstein 2006]. In both cases the derivation contains a mathematical mistake: in Case A adding equations with different domains, in Case B changing the physical problem (problem substitution) presented with the word "generally" as if it were a logical consequence. This revised version (October 2026) clarifies the original argument from May 2017 in response to the comment by V.G. Rousseau (2018).