  Anyons with arbitrary exchange phases exist on 1D lattices in ultracold
gases. Yet, known continuum theories in 1D do not match. We derive the
continuum limit of 1D lattice anyons via interacting bosons. The theory
maintains the exchange phase periodicity fully analogous to 2D anyons. This
provides a mapping between experiments, lattice anyons, and continuum theories,
including Kundu anyons with a natural regularization as a special case. We
numerically estimate the Luttinger parameter as a function of the exchange
angle to characterize long-range signatures of the theory and predict different
velocities for left- and right-moving collective excitations.
