Math 2.0

Real functions Proposition

C^1 functions are strictly differentiable

Builds on Strict differentiabilityC^k function characterised by continuous derivativesMean value inequality

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let f ⁣:U→ℝmf \colon U \to \mathbb{R}^m be of class C^1 on UU. Then ff is strictly differentiable at each point a∈Ua \in U, with linear map Df(a)Df(a).

Remarks

Proof: let ε>0\varepsilon > 0 and choose r>0r > 0 with ∥Df(z)−Df(a)∥≤ε\|Df(z) - Df(a)\| \le \varepsilon for ∥z−a∥<r\|z - a\| < r. Apply the mean value inequality to g(z)=f(z)−Df(a)zg(z) = f(z) - Df(a) z on the ball of radius rr. This gives ∥f(x)−f(y)−Df(a)(x−y)∥≤ε∥x−y∥\|f(x) - f(y) - Df(a)(x - y)\| \le \varepsilon \|x - y\| for x,yx, y in that ball. For a function of one variable on an open interval, strict differentiability at each point is equivalent to being of class C1C^1.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionMean value inequality, statement, no evidence levelMean value inequalitystatementStrict differentiability, definition, no evidence levelStrict differentiabil…definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementC^1 functions are strictly differentiable, statement, no evidence levelC^1 functions are str…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Limit of a function of several variables concept
  6. Differentiable function of several variables definition
  7. Partial derivative definition
  8. Differentiable on an open set definition
  9. Mean value inequality statement
  10. Strict differentiability definition
  11. Higher partial derivatives definition
  12. Iterated derivative definition
  13. C^k function characterised by continuous derivatives statement
  14. C^1 functions are strictly differentiable statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Strict_differentiability, Basic definition
About this page
Tag0STPPDXGME
URIm2://0STPPDXGME
Aliasescontinuously-differentiable-implies-strictly-differentiable
Kindstatement, proposition
WritingW0 stub
MSC26B05
StatusThis copy of the site has no build data. Run m2 status 0STPPDXGME.
UsesStrict differentiability, C^k function characterised by continuous derivatives, Mean value inequality

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