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

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A continuous function <math>f : I \to X</math> from a non-empty and 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}}''' in <math>X.</math>

A '''{{em|[[Path (topology)|path]]}}''' in <math>X</math> is a curve in <math>X</math> whose ___domain is compact while an '''{{em|[[Arc (mathematicstopology)|arc]]}}''' or '''{{em|{{mvar|C}}⁰-arc}}''' in <math>X</math> is a path in <math>X</math> that is also a [[topological embedding]].

For any <math>k \in \{ 1, 2, \ldots, \infty \},</math> a curve <math>f : I \to X</math> valued in a topological vector space <math>X</math> is called a '''{{em|<math>C^k</math>-embedding }}''' if it is a [[topological embedding]] and a <math>C^k</math> curve such that <math>f^{\prime}(t) \neq 0</math> for every <math>t \in I,</math> where it is called a '''{{em|<math>C^k</math>-arc}}''' if it is also a path (or equivalently, also a <math>C^0</math>-arc) in addition to being a <math>C^k</math>-embedding.