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How to understand the following definition of curve $\gamma:[a,b]\to\mathbb R^2$?

The above means it's a function that assigns a pair of numbers to every pair of numbers. But isn't a curve actually a collection of vectors in $\mathbb{R}^2$? I don't understand how it can be expressed as a function.

user216094
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    It assigns a pair of numbers to a real number, not to a pair of real numbers. More specifically: it assigns a pair of real numbers to every real number in the interval $[a,b]$. So (more or less) yes: the image of this function is a collection of elements of $\mathbb{R}^2$, which forms the graphical representation of the curve. – StackTD Oct 07 '15 at 14:43

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