Power factor correction AC motor

Apparently you have a inductive load to a AC power supply, and want to cancel the reactive part of the load by putting a capacitor accross it. You mention values Rs and Rl, but since you provided no definition for these, we can't tell how they are hooked up and therefore their relevance.

However, if I understand the problem correctly, this is mostly about finding the capacitance that cancels the inductance. From basic circuits you know that the impedance magnitude of a capacitance is 1/ωC, and for a inductance is ωL. Setting these two equal and solving for C yields C = 1/ω²L = 1/(2πf)²L = 880 µF.

My experience with PFC supplies is mostly theoretical, but as general comments about SMPS, I can say a few things:

Continuous mode sometimes have "prettier" waveforms. This might translate to lower harmonics, if you care. That's why discontinuous mode requires better filtering.

"Correct" selection of modes may be less of a concern. The bigger concern is often staying in one mode, because your controls are all designed around it. Mode transitions tend to be more difficult for controllers to handle.

Boost converters (which share a lot with PFC supplies as I understand them) will enter discontinuous mode at low loads. Which implies that a well-designed continuous-mode controller can probably handle discontinuous mode as well. I'd be surprised if anyone marketed a controller that failed at open load!

If it was me, I'd try the average current continuous mode control (which you seem to be interested in). I'd be surprised if it didn't do what you want. The tradeoffs seem mostly to be about cost in components, which seems to not concern you.