Math 2.0

Real functions Definition

Continuously differentiable function

Builds on Differentiable functionContinuous functionHigher derivative

Definition

Let II be an open interval and let f ⁣:I→ℝf \colon I \to \mathbb{R}. The function ff is continuously differentiable, or of class C1C^1, if ff is differentiable on II and f′f' is continuous on II. More generally, for k≥1k \ge 1, ff is of class CkC^k if its higher derivative f(k)f^{(k)} exists and is continuous on II. The class C0C^0 consists of the continuous functions.

Remarks

Each Ck+1C^{k+1} function is CkC^k. A differentiable function need not be C1C^1. See x^2 sin(1/x). The function x∣x∣x |x| is C1C^1 but not twice differentiable at 00. Mathlib defines the class as ContDiff ℝ k f.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionDerivative function, definition, no evidence levelDerivative functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionContinuously differentiable function, definition, no evidence levelContinuously differen…definition
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Continuity at a point definition
  4. Derivative definition
  5. Continuous function definition
  6. Differentiable function definition
  7. Derivative function definition
  8. Higher derivative definition
  9. Continuously differentiable function definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Smoothness, Differentiability classes
About this page
Tag06SKCHDQT5
URIm2://06SKCHDQT5
Aliasesc1-function, ck-function
Kinddefinition
WritingW1 stated
MSC26A24
DefinesContinuously differentiable function
UsesDifferentiable function, Continuous function, Higher derivative

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