Discovering 'the mainland' - How?

In that film about Columbus (the better, Ridley Scott one) they mention how he never got to explore further and find the mainland. And it wasn’t until the late 1800s that Antarctica was found to be a continent rather than just a group of islands.

My question is, lacking a satellite view, how and when exactly do you know that something is a continent and not just a really big island?

When you circumnavigate it.

Actually, on his third voyage he explored the coast of what’s now Venezuela and the fourth covered quite a stretch of central America. What he never reached was the mainland of what became the United States.

In Columbus’ case, he realised that he was dealing with something more than offshore islands when he encountered the mouth of the Orinoco River. Huge freshwater estuaries indicated some sort of giant river and hence some huge hinterland as a catchment area. (Then again, this was during his particularly bonkers stage when he was comparing the Earth to a woman’s breast etc., so his grasp of matters may not have been too lucid.)
Exapno’s criterion isn’t quite sufficient: Antarctica had been circumnavigated long before people were even sure it was there.

Are you referring to Cook’s voyages? He’s not normally credited with ever seeing the Antarctic mainland. That rather puts his travels outside the context of the OP.

Fabian Gottlieb von Bellingshausen is credited with being the first to discover the mainland and the first to sail around it, thereby becoming the first to realize what it was.

If you don’t mean Cook, then I need more info about who did a circumnavigation when.

Actually, the slightly flippant example I had in mind was Magellan’s crew. The capes extend sufficiently far south that any voyage round the world is effectively a circling of Antarctica. That trip had consequences for the possible geography of any southern landmass, if there were one, since that’s the point when it became obvious it had to be disconnected from the known continents.