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

Revision as of 23:58, 14 February 2021

Revision as of 00:02, 15 February 2021

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where <math>p = \left(p_1, \ldots, p_n\right).</math>

If <math>f</math> is differentiable at a point then it is continuous at that point.{{sfn|Trèves|2006|pp=412–419}}

If <math>f</math> is differentiable at every point in some subset <math>S</math> of its ___domain then <math>f</math> is said to be ('''{{em|once}}''' or '''{{em|<math>1</math>-time}}''') '''{{em|differentiable in <math>S</math>}}''', where if the subset <math>S</math> is not mentioned then it is this means that it is differentiable at every point in its ___domain.

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>