Discretely synchronized, distributed real-time systems may su er from a time discontinuity problem in that local clocks observe the disappearance or reappearance of time intervals. This problem occurs since traditional discrete clock synchronization algorithms adjust local clocks instantaneously. Such time discontinuities may lead to run-time faults due to the loss or gain of critical time points such as task release times and deadlines. In this paper we propose a dynamic constraint transformation technique we call a constraint transformation for equi-continuity (CTEC), to correctly enforce timing requirements in a distributed real-time system possessing periodically synchronized distributed local clocks. While continuous clock synchronization is generally suggested to avoid the time discontinuity problem, it incurs too much run-time overhead to be implemented in software. The proposed CTEC technique can solve the time discontinuity problem without modifying discrete clock synchronization algorithms. The CTEC working as an added component of discrete clock synchronization moves timing constraints out of discontinuous time intervals. In doing so, it makes use of a mapping derived from continuous clock synchronization in order to exploit its continuity property. We formally prove the correctness of CTEC by showing that the CTEC with discrete clock synchronization generates the same task schedule as continuous clock synchronization. In order to show the e ectiveness of CTEC, we have implemented it on a distributed platform based on the CAN bus, and performed extensive experiments. The experimental results indicate that time discontinuities present a consistency problem to real-world systems. They also show that CTEC is an e ective solution to the problem, while incurring little run-tine overhead.