  We study the effects of quenched disorder on the commensurate-incommensurate
transitions in the 1D $\mathbb{Z}_N$ chiral clock model. The interplay of
domain walls and rare regions rounds the sharp transitions of the pure model.
The density of domain walls displays an essential singularity, while the order
parameter develops a discontinuity at the transition. We perform extensive
density-matrix renormalization group calculations to support theoretical
predictions. Our results provide a distinct rounding mechanism of continuous
phase transitions in disordered systems.
