Homogeneous sliding-mode-based differentiators (HD) are known for their high asymptotic accuracy. Their practical realization is computer-based and requires discretization. The corresponding combination of a discrete system with a continuous-time input signal produces hybrid dynamics. In the case of the most usual one-step Euler discretization that hybrid system lacks the homogeneity of its predecessor and loses its ultimate accuracy. Nevertheless, the discrete differentiator can be modified, restoring the homogeneity and the accuracy of HD. Similarly to HD, the proposed homogeneous discrete differentiator can also be used to differentiate signals with a variable upper bound of the highest derivative. Simulation results confirm the theoretical results.