Math 2.0

definition

Differentiable function

Definition

Let II be an open interval and let f ⁣:I→Rf \colon I \to \mathbb{R} be a function. The function ff is differentiable at a∈Ia \in I if ff has a derivative at aa. It is differentiable on II if it is differentiable at each point of II.

Remarks

Differentiable functions are continuous. The converse fails: the absolute value is continuous but not differentiable at 00. Polynomials, the exponential function, and the sine are differentiable on R\mathbb{R}. Mathlib defines differentiability at a point as DifferentiableAt.

Explanations

L1 human

A differentiable function has a well defined slope at each point. Its graph is smooth, with no corners and no vertical tangents.

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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 functiondefinitionDerivative, definition, no evidence levelDerivativedefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definition
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Derivative definition
  4. Differentiable function definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Differentiable_function, Differentiability of real functions of one variable

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