  We prove that the local version of the chain rule cannot hold for the
fractional variation defined in arXiv:1809.08575. In the case $n = 1$, we prove
a stronger result, exhibiting a function $f \in BV^{\alpha}(\mathbb{R})$ such
that $|f| \notin BV^{\alpha}(\mathbb{R})$. The failure of the local chain rule
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
results of arXiv:2111.13942 and the distributional approach of the previous
papers arXiv:1809.08575, arXiv:1910.13419, arXiv:2011.03928, arXiv:2109.15263.
As a byproduct, we refine the fractional Hardy inequality obtained in
arXiv:1611.07204, arXiv:1806.07588 and we prove a fractional version of the
closely related Meyers-Ziemer trace inequality.
