We prove that the local version of the chain rule cannot hold for the fractional variation defined in our previous article (2019). In the case n = 1, we prove a stronger result, is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the distributional techniques developed in our previous works (2019-2022). As a byproduct, we refine the fractional Hardy inequality obtained in works of Shieh and Spector (2018) and Spector (J. Funct. Anal. 279 (2020), article no. 108559) and we prove a fractional version of the closely related Meyers-Ziemer trace inequality.
