which is a piecewise-defined function with a linear "critical piece". this function has the property that it is continuous for all real "z" (formula_7 continuous). the first authors to recognize this approximation were rubstov and romerio, although it is not in their paper, it can be found in their algorithm that is used in their software prototype. the linear approximation to tetration, on the other hand, had been known before, for example by ioannis galidakis. this is a natural inverse of the linear approximation to tetration.