Pointwise limit distribution results are given for the isotonic regression estimator at a point of discontinuity. The cases treated are independent data, - and -mixing data and subordinated Gaussian long range dependent data. Pointwise limit results for the nonparametric maximum likelihood estimator of a monotone density are given at a point of discontinuity, for independent data. The limit distributions are non-standard and differ from the ones obtained for differentiable regression and density functions.