Existence of inverse function in a neighbourhood of a critical value

The classical inverse function theorems guarantee the existence of an inverse function in a neighborhood of the value of a given point if the regularity condition is satisfied at this point, that is, the first derivative at a given point is nondegenerate. A more general condition for the existence of an implicit function is the 2-regularity condition. It holds, for example, for many quadratic mappings at zero. It is known that under natural smoothness assumptions, the existence of a continuous inverse function follows from a 2-regularity of a map at a point in a certain direction. In this paper, it is shown that, in the known statements guaranteeing the existence of an inverse function when the 2-regularity condition is satisfied, we can weaken the smoothness assumptions. However, the inverse function may not be continuous.