Fun fact, this actually depends on who you're asking. There are several different ways of counting them, and different countries count them differently. For example, some count North and South America as one continent, some have Europe and Africa as one, and some have all three Europe, Africa, and Asia counted as one continent.
In geography at school i was always told that Australia and Papua new Guinea were a continent called Australasia and that Oceania is not.
I'm sure that there are plenty of other opinions on that I am sure.
I have always been told there were seven, though i would be ready (as a European myself) to accept that Europe doesn't really count basically being more of a subcontinent. But i don't think that i could accept three as a valid answer.
When I was growing up they just said "Australia is a continent", they never specified about the islands around it or whether they were referring to the continent as Australia or simply saying that the island of Australia is a continent in and of itself. Personally I refer to the continent as Oceania and include the relevant islands along with Australia
I'd go with geological rulings, personally. Greenland is part of Europe, but technically, it's on the north American plate. It is way more part of NA than it is Europe, not that it matters. But in two million years, Greenland will still be as close as it is to NA, while it will have drifted away from Europe. In the meantime, other tectonic plates, like India's, will still be pushing into the rest of Asia. If south America drifted away from North America, we'd be forced to consider them both separate. But if they stick together, they should only be counted as one. Anyway,in my heart, there will always only be one true continent: Pangaea!