In this article, we investigate regular curves whose derivatives have vanishing mean oscillations. We show that smoothing these curves using a standard mollifier one gets
regular curves again. We apply this result to solve a couple of open problems. We show that curves with finite Möbius energy can be approximated by smooth curves in the energy space W 3 2,2 such that the energy converges which answers a question of He. Furthermore, we prove conjectures by Ishizeki and Nagasawa on certain parts of a decomposition of the Möbius energy and extend a theorem of Wu on inscribed polygons to curves with derivatives with vanishing mean oscillation. Finally, we show that the result by Scholtes on the -convergence of the discrete Möbius energies towards the Möbius energy also holds for curves of merely bounded energy.