Math 2.0

Real functions Definition

Smooth function

Builds on Continuously differentiable function

Definition

Let II be an open interval and let f ⁣:I→ℝf \colon I \to \mathbb{R}. The function ff is smooth, or of class C∞C^{\infty}, if it is of class CkC^k for each natural number kk.

Remarks

So a smooth function has derivatives of all orders. Polynomials, the exponential function, and the sine are smooth on ℝ\mathbb{R}. A smooth function need not be analytic. See the smooth non-analytic function. Mathlib uses ContDiff ℝ ⊤ 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…definitionSmooth function, definition, no evidence levelSmooth functiondefinition
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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
  10. Smooth function definition

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18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Smoothness, Differentiability classes
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