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

Revision as of 21:31, 14 February 2021

Revision as of 21:32, 14 February 2021

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If <math>f : I \to X</math> is differentiable then it is said to be '''{{em|continuously differentiable}}''' or '''{{em|<math>C^1</math>}}''' if its '''{{em|derivative}}''', which is the induced map <math>f^{\prime} = f^{(1)} : I \to X,</math> is continuous.

Using induction on <math>1 < k \in \mathbb{N},</math> the map <math>f : I \to X</math> is '''{{em|<math>k</math>-times continuously differentiable}}''' or '''{{em|<math>C^k</math>}}''' if its <math>k-1^{\text{th}}</math> derivative <math>f^{(k-1)} : I \to X</math> is continuously differentiable, in which case the '''{{em|<math>k^{\text{th}}</math>-derivative of <math>f</math>}}''' is the map <math>f^{(k)} := \left(f^{(k-1)}\right)^{\prime} : I \to X.</math>

It is called '''{{em|smooth}}''' or '''{{em|infinitely differentiable}}''' if it is <math>k</math>-times continuously differentiable for every integer <math>k \in \N.</math>

For <math>k \in \mathbb{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-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}}'''.