If a person was born in North America, would they be called North American? because that is the continent were they born like Europeans are called because they born in Europe?

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Yes, if you were born in North America, you could call yourself a North American.

Explanation:

Many people born in North America identify themselves with their country rather than their continent. They would say "I'm a Canadian", "I'm American", or "I'm from Mexico" when asked where they are from. However, it is still correct to call yourself a North American if you are born in this continent.

where is the map at?