Scale free SL(2,R) analysis and the Picard's existence and uniqueness theorem

The existence of higher derivative discontinuous solutions to a first order ordinary differential equation is shown to reveal a nonlinear SL(2,R) structure of analysis in the sense that a real variable $t$ can now accomplish changes not only by linear translations $t \to t + h$ but also by inversions $t \to 1/t$. We show that the real number set has the structure of a positive Lebesgue measure Cantor set. We also present an extension of the Picard's theorem in this new light.