Intrinsic Supermoothness

Abstract

The phenomenon, known as "supersmoothness" was first observed for bivariate splines and attributed to the polynomial nature of splines. Using only standard tools from multivatiate calculus, we show that if we continuously glue two smooth functions along a curve with a "corner", the resulting continuous function must be differentiable at the corner, as if to compensate for the singularity of the curve. Moreover, locally, this property, we call supersmoothness, characterizes non-smooth curves. We also generalize this phenomenon to higher order derivatives. In particular, this shows that supersmoothness has little to do with properties of polynomials.