Differentiable vector-valued functions from Euclidean space: Difference between revisions
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== Continuously differentiable vector-valued functions ==
A map <math>f,</math> which may also be denoted by <math>f^{(0)},</math> between two [[topological space]]s is said to be '''{{em|<math>0</math>-times continuously differentiable}}''' or '''{{em|<math>C^0</math>}}''' if it is continuous. ▼
=== Curves ===
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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 map is said to be '''{{em|<math>0</math>-times continuously differentiable}}''' or '''{{em|<math>C^0</math>}}''' if it is continuous.
A continuous 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:
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