"Average" refers to filtering the inductor current waveform so that the control behaves as a linear circuit without switching (or subtracting the triangle waveform from the baseline, to the same end).
Average mode control allows smaller ripple fractions and greater current limit accuracy than slope compensated peak current mode does. This makes it particularly useful for low ripple applications, where you might save significant cost on the inductor by using a shitty powdered iron type.
This works whether current is continuous or not, because several cycles are averaged over.
Tim