Given a discrete-time uncertain single-inputsingle-output (SISO) system, matched uncertainties, and a measured output, a necessary and sufficient condition is given for the existence of a min-max output feedback stabilizing controller. The existence problem of an output stabilizing controller for uncertain systems was treated extensively and solved for the continuous case, but it remained open for the discrete-time case. It turns out that in contrast to the continuous case, the discrete-time system is required to be neither discrete strictly positive real nor even minimum phase and still may satisfy the existence conditions. The paper also contains some new results on discrete positive real functions.