We consider a simple mean reverting diffusion process, with piecewise constant drift and diffusion coefficients, discontinuous at a fixed threshold. We discuss estimation of drift and diffusion parameters from discrete observations of the process, with a generalized moment estimator and a maximum likelihood estimator. We develop the asymptotic theory of the estimators when the time horizon of the observations goes to infinity, considering both cases of a fixed time lag (low frequency) and a vanishing time lag (high frequency) between consecutive observations. In the setting of low frequency observations and infinite time horizon we also study the convergence of three local time estimators, that are already known to converge to the local time in the setting of high frequency observations and fixed time horizon. We find that these estimators can behave differently, depending on the assumptions on the time lag between observations.