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[MistakeNot]

"America" (as I understand it from over here in Australia) is specifically a shortened form of "the United States of America", a country within the continent of North America.

There is no single continent called America, and I would understand any reference to "America" to be equal to "USA".

Depends.
"America" is also used as a term for North America and South America together, and that was also the original meaning of the name.
Also, in many places America as a whole is actually considered to be a single continent. In other places it is considered to be composed of two continents - North and South America.