Differentiable vector-valued functions from Euclidean space: Difference between revisions
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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 \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}}''', <math>C^\infty,</math> 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 \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.
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If <math>f</math> is differentiable and if each of its partial derivatives is a continuous function then <math>f</math> is said to be ('''{{em|once}}''' or '''{{em|<math>1</math>-time}}''') '''{{em|continuously differentiable}}''' or '''{{em|<math>C^1.</math>}}'''{{sfn|Trèves|2006|pp=412–419}}
For <math>k \in \N,</math> having defined what it means for a function <math>f</math> to be <math>C^k</math> (or <math>k</math> times continuously differentiable), say that <math>f</math> is '''{{em|<math>k + 1</math> times continuously differentiable}}''' or that '''{{em|<math>f</math> is <math>C^{k+1}</math>}}''' if <math>f</math> is continuously differentiable and each of its partial derivatives is <math>C^k.</math>
Say that <math>f</math> is <math>C^{\infty},</math> '''{{em|smooth}}''', <math>C^\infty,</math> or '''{{em|infinitely differentiable}}''' if <math>f</math> is <math>C^k</math> for all <math>k = 0, 1, \ldots.</math>
The '''{{em|[[Support (mathematics)|support]]}}''' of a function <math>f</math> is the [[Closure (topology)|closure]] (taken in its ___domain <math>\operatorname{Dom} f</math>) of the set <math>\{ x \in \operatorname{Dom} f : f(x) \neq 0 \}.</math>
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