I am unclear how the "discontinuities" in the waveform at the junction of each data point can be physically produced?
They can't, exactly. In the real world you may come close, but everything is a low-pass process, so even if you "order" the electronics or whatever to make a sharp discontinuity, the actual generated waveform will be continuous, with a "sharpness" dependent on bandwidth.
The way you approach this, in 2022, is to generate the waveform digitally and then apply it to a digital to analog converter. The way you would have approached this in 1975 would have been to generate a sine wave in analog circuitry that had inphase and quadrature components available, then, depending on the code, switch in the inphase, quadrature, negative inphase, or negative quadrature.
Note: in actual practice you want to bandpass filter this, but there's a science to getting that filtering right. For now, concentrate on understanding the theory; when you're ready you can start learning about proper filtering.
Taking a speaker as an example, how is it possible to produce this "discontinuity" in the sound wave by using the speaker cone's mechanism?
Use the highest bandwidth speaker you can, and feed it with the highest-bandwidth version of the QAM signal that you can. Or, bandpass filter the signal properly, and feed it to the (ultimately, much happier) speaker.