Differentiable vector-valued functions from Euclidean space: Difference between revisions

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=== Curves ===

Differentiable curves are an important special case of differentiable vector-valued (i.e. TVS-valued) functions which, in particular, are used in the definition of the [[Gâteaux derivative]]. They are fundamental to the analysis of maps between two arbitrary [[topological vector space]]s <math>X \to Y</math> and so also to the analysis of TVS-valued mapmaps from a [[Euclidean space]]s, which is the focus of this article.

A continuous function <math>f : I \to X</math> from a non-degenerate interval <math>I \subseteq \R</math> into a [[topological space]] <math>X</math> is called a '''{{em|curve}}''',or a '''{{em|<math>C^0</math> curve}}''', and it is also said to be '''{{em|<math>0</math>-times continuously differentiable}}'''.