  In this paper, we consider a class of continuous maps characterized by a
singularity of order $x^{q/p}$ (with $p,q \in \mathbb{N}$, $p>q$, and
$(p,q)=1$) on one side of the discontinuity boundary $\Sigma$ and a linear
behaviour on the other side. Such maps arise naturally in the study of grazing
bifurcations of hybrid and piecewise flows. In this context the boundary
collision of a fixed point of the map with $\Sigma$ then corresponds to a
grazing bifurcation of the flow. We will start by studying one-dimensional
maps, and the main result of this paper is a classification of all bifurcation
scenarios, including: period doubling and robust chaos.
