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

Revision as of 23:06, 14 February 2021

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For <math>k \in \N,</math> it is called '''{{em|<math>k</math>-times differentiable}}''' if it is <math>k-1</math>-times continuous differentiable and <math>f^{(k-1)} : I \to X</math> is differentiable.

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}}'''.

A '''{{em|[[Path (topology)|path]]}}''' in <math>X</math> is a curve in <math>X</math> whose ___domain is compact while an '''{{em|[[Arc (mathematics)|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> is called a '''{{em|<math>C^k</math>-arc}}''' or 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> andwhere <math>fit :is Icalled \toa X'''{{em|<math>C^k</math>-arc}}''' if it is ana path (or equivalently, a <math>C^0</math>-arc) (i.e.in aaddition [[topologicalto being a <math>C^k</math>-embedding]]).

=== Differentiability on Euclidean space ===