In this paper, we show that amplitude transitions that are inherent to the multilevel outphasing radio transmitter architecture distort the transmitted signal due to time-domain discontinuities. In order to address this challenge, we propose a new transmitter architecture called tri-phasing which avoids discontinuities in signal waveforms and thus achieves significantly better linearity than multilevel outphasing. The output waveform in tri-phasing can be made continuous by representing the baseband signal with three components. One of the three components is amplified by discrete amplitude steps, whereas the other two are used to compensate the instantaneous shift in the output waveform due to the discrete amplitude step and to provide fine amplitude resolution. An implementation of the tri-phasing transmitter requires three phase modulators and additional digital signal processing. The system-level simulations performed in this paper demonstrate that the ACLR of a multilevel outphasing transmitter with 4 amplitude levels and 10bit phase resolution is limited to-48 dBc, when simulated with a 100 MHz carrier-aggregated LTE downlink signal at 2.46 GHz carrier frequency. The proposed tri-phasing transmitter achieves-58 dBc ACLR with the same simulation parameters, indicating that continuous amplitude transitions can significantly improve the transmitter linearity.