  We investigate the 1D Riemann-Liouville fractional derivative focusing on the
connections with fractional Sobolev spaces, the space $BV$ of functions of
bounded variation, whose derivatives are not functions but measures and the
space $SBV$, say the space of bounded variation functions whose derivative has
no Cantor part. We prove that $SBV$ is included in $W^{s,1} $ for every $s \in
(0,1)$ while the result remains open for $BV$. We study examples and address
open questions.
