  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.
