Actually, "America" isn't a Continent. "The Americas" is consisted of 3 separate continents, North, Central, and South America. However, everyone just calls it "America" to refer to all three continents.
Technically speaking, the United States of America should actually be named "The United States of North America" or "USNA", since the United States is a part of North America, along with Canada, Greenland, and Mexico.
Central America would consist of Belize, Costa Rica, El Salvador, Guatemala, Honduras, Nicaragua, and Panama.
South America consists of Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Guyana, Paraguay, Peru, Suriname, Uruguay, and Venezuela.
The reason that most people refer to The United States as "America" is simply because The United States made itself known to the world more than the other countries did. Think of it this way... You have a Team of 10 Moderators. The whole Moderator Team represents a Continent, however, each Moderator represents a different Country in that Continent. Out of the 10 Moderators, one of them stands out the most, and makes a bigger impact than the rest. That Moderator then receives a Promotion, and becomes an Administrator. So now, that Administrator will forever go down in history, whether it be a good history, or a bad one. <- That's the best way I can relate the USA to RSPS terms.
Most schools around the world teach the History of their Country, and then when they finish that, they usually teach World History, and/or the History of the United States. The reason why is because The United States is probably one of, if not the most iconic Country in History, and no, I'm not just saying that because I'm from the United States. The United States is, I believe, the youngest recognizable Country in the world, as the United States has only been an established country for just under 239 years, yet, over those 239 years, we've successfully grown the Strongest Army in the World, and have grown to have the #1 Economy in the world (Up until China Recently surpassed us), and we have been the determining factor in most major wars, most specifically, World Wars 1 and 2.
However, I can understand where you're coming from. It's like the same thing when people refer to The Middle East. The first country or countries they think of is Iraq and/or Afghanistan, or they refer to Afghanistan as the Middle East.