I've had the chance to drive a number of CVT cars. My own car is a Subaru Outback, and I've previously rented a Nissan Murano, Rav4 Hybrid, and Corolla, all with CVTs as well.
One interesting characteristic I've noticed across these cars is that they seem to index the transmission on gear numbers. As an example, if I'm accelerating from a stop, the car will rev up as it accelerates, then heave as if it were going into neutral for a gear change, then continue to accelerate starting at lower revs, just like what I'd expect in non-CVT automatic designs, as well as manual transmissions.
I understand that for many of these cars, indexed gears make sense if you put the shifter into the manumatic setting (usually a managed sequential mode of operation; AKA Tiptronic, Sportshift, Shiftronic, etc., depending on the car brand), but from my reading, many CVT designs can continue to provide power even as the gear ratios change.
Is it the case that more common CVT designs can't provide power while changing gear ratios? If not, why would a modern CVT car not accelerate continuously?