Continuous Phase Modulation
What is important is the envelope of the modulated signal. For M-PSK, due to the phase discontinuities, the envelope droops every time there is a phase transition. The envelope even goes to zero if there is a phase transition of $180$ degrees. So M-PSK does not have a constant envelope.

Changes in the envelope are largely reduced by using offset QPSK (OQPSK), and they are completely eliminated by using continuous-phase modulation schemes, such as minimum-shift keying (MSK), which is a special case of OQPSK.

Take a look at these related questions and its answers: Q1, Q2, Q3.