  The conjugacy problem is one of the central questions in iteration theory. As
far as we, for discontinuous strictly monotone maps there is no complete
result. In this paper, we investigate the conjugacy problem of strictly
monotone maps with only one jump discontinuity. We give some sufficient and
necessary conditions for the conjugacy relationship. And we present some
methods to construct all conjugacies. Furthermore, we present the conditions to
guarantee $C^1$ smoothness of these conjugacies.
