Time transfer (TT) is the process of transmitting a timing signal from one place to another place. It has applications to the formation and realization of Coordinated Universal Time (UTC), telecommunications, electrical power grids, and even stock exchanges. TT is the actual bottleneck of the UTC formation and realization since the technology of atomic clocks is almost always ahead of that of TT. GPS carrier-phase time transfer (GPSCPTT), as a mainstream TT technique accepted by most national timing laboratories, has suffered from the day-boundary-discontinuity (day-BD) problem for many years. This makes us difficult to observe a remote Cesium fountain clock behavior even after a few days. We find that day-BD comes from the GPS code noise. The day-BD can be lowered by ~40% if more satellite-clock information is provided and if a few GPS receivers at the same station are averaged. To completely eliminate day-BD, the RINEX-Shift (RS) algorithm has been designed. The RS result matches the two-way satellite time/frequency transfer (TWSTFT) result much better than the conventional GPSCPTT result. With the RS algorithm, we are able to observe a remote Cesium fountain after half a day. We also study the BD due to GPS data anomalies (anomaly-BD). A simple curve-fitting strategy can eliminate the anomaly-BD. Thus, we achieve continuous GPSCPTT after eliminating both day-BD and anomaly-BD.