Finite distortion curves: Continuity, Differentiability and Lusin's (N) property

We define finite distortion $\omega$-curves and we show that for some forms $\omega$ and when the distortion function is sufficiently exponentially integrable the map is continuous, differentiable almost everywhere and has Lusin's (N) property. This is achieved through some higher integrability results about finite distortion $\omega$-curves. It is also shown that this result is sharp both for continuity and for Lusin's (N) property. We also show that if we assume weak monotonicity for the coordinates of a finite distortion $\omega$-curve we obtain continuity.