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

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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 maps from [[Euclidean space]]s, which is the focus of this article.

A continuousmap functionis <math>fsaid : I \to X</math> from a non-degenerate intervalbe '''{{em|<math>I \subseteq \R0</math>-times intocontinuously a [[topological space]] <math>X</math> is called a '''{{em|curvedifferentiable}}''',or aor '''{{em|<math>C^0</math> curve}}''', andif it is also said to be '''{{em|<math>0</math>-times continuously differentiable}}'''continuous.

A curvecontinuous map <math>f : I \to X</math> from a subset <math>I \subseteq \mathbb{R}</math> that is valued in a [[topological vector space]] <math>(X, \tau)</math> is said to be ('''{{em|once}}''' or '''{{em|<math>1</math>-time}}''') '''{{em|differentiable}}''' if for all <math>t \in I,</math> it is '''{{em|differentiable at <math>t,</math>}}''' which by definition means the following limit in <math>(X, \tau)</math> exists:

:<math>f^{\prime}(t) := f^{(1)}(t)

:= \lim_{\stackrel{r \to t}{t \neq r \in I}} \frac{f(r) - f(t)}{r - t}

= \lim_{\stackrel{0 \neq h \to 0}{t \neq t + h \in I}} \frac{f(t + h) - f(t)}{h}.</math>

A differentiable curveIf <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> athe curvemap <math>f : I \to X</math> is '''{{em|<math>k</math>-times continuously differentiable}}''' 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>

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

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}}'''.

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 <math>C^k</math> curve, <math>f^{\prime}(t) \neq 0</math> for every <math>t \in I,</math> and <math>f : I \to X</math> is an <math>C^0</math>-arc (i.e. a [[topological embedding]]).