I know that a curve is a continuous function from the unit interval to $\mathbb{R}^n$. That raises the question, can a single point be considered the image of a curve?
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1The constant function has the single point on which every number is mapped as the complete image. Hence, the answer is "yes". – Peter May 21 '21 at 16:34
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Take also into account that the word "curve" is reused to name multiple concepts. When distinction is needed, some people call the concept that you defined "parametric curve". – plop May 21 '21 at 16:42