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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  ∈ (0, 1) while the result remains open for  BV . We study examples and address open questions.

Fractional sobolev spaces and functions of bounded variation of one variable