  We prove that if a curve parametrized by arc length is a stationary point of
the Moebius energy introduced by Jun O'Hara, then it is smooth whenever the
Moebius energy is finite. Our methods, interestingly, only rely on purely
analytical arguments, entirely without using Moebius invariance. Furthermore,
the techniques involved are not fundamentally restricted to one-dimensional
domains, but are generalizable to arbitrary dimensions.
