A vector-valued function is a function with values in a vector space. For finite-dimensional real or complex target spaces, choosing a basis represents it by scalar coordinate functions; continuity and differentiability can then be checked coordinatewise.
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A vector-valued function is a function that takes one or more variables (often real numbers) as input and outputs a vector. In other words, instead of producing a single scalar value for each input, a vector-valued function yields a vector, which is an ordered collection of numbers. These vectors often represent quantities that have both magnitude and direction.