  The Skorokhod reflection was used in 1961 to create a reflected diffusion on
the half-line. Later, it was used for processes with jumps such as reflected
L\'evy processes. Like a Brownian motion, which is a weak limit of random
walks, reflected processes on the half-line serve as weak limits of random
walks with switching regimes at zero: one regime away from zero, the other
around zero. In this article, we develop a general theory of this regime change
and prove convergence to a function with generalized reflection. Our results
are deterministic and can be applied to a wide class of stochastic processes.
Applications include storage processes, heavy traffic limits, diffusion on a
half-line with a combination of continuous reflection, jump exit, and a delay
at 0.
