Differentiable function
Builds on Derivative
Definition
Let be an open interval and let be a function. The function is differentiable at if has a derivative at . It is differentiable on if it is differentiable at each point of .
Remarks
Differentiable functions are continuous.
The converse fails: the absolute value is continuous but not differentiable at .
Polynomials, the exponential function, and the sine are differentiable on .
Mathlib defines differentiability at a point as DifferentiableAt.
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
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- Limit point of a set definition
- Limit of a function definition
- Derivative definition
- Differentiable function definition
Where this leads
84 pages build on this one.
- Taylor's theorem with the Cauchy remainder
- 1 + x is at most exp x
- Reduction formula
- Weierstrass function
- sin x / x tends to 1
- Taylor series
- Derivative function
- Power series of sine
- Taylor's theorem with the Peano remainder
- Taylor's theorem
- Maclaurin series
- log x is at most x - 1
- Derivative of arctangent
- Power series of the exponential function
- Continuously differentiable function
- Derivatives of the hyperbolic functions
- Differentiable functions are continuous
- Higher derivative
- Power series of cosine
- Integration by parts
- Smooth non-analytic function
- Smooth function
- Quotient rule
- Absolute value is continuous but not differentiable at 0
- Integration by substitution
- The exponential function is convex
- Trigonometric substitution
- The gamma function extends the factorial
- Second derivative test
- Dirichlet integral
- Taylor's theorem with the Lagrange remainder
- Taylor polynomial
- Power rule
- General Leibniz rule
- Stirling's formula
- Derivative of a polynomial
- Integral of a power
- Reciprocal rule
- pi is greater than 3
- A continuous nowhere differentiable function
- Derivative of tangent
- Power series of arctangent
- Product rule
- Improper integral of x^p on an unbounded interval
- Tangent half-angle substitution
- x^2 sin(1/x) is differentiable with discontinuous derivative
- Taylor's theorem with the integral remainder
- Wallis product
- Second derivative
- A function with nonnegative second derivative is convex
- Laplace's method
- Lebesgue's theorem on differentiability of monotone functions
- Minimal curvature property of natural cubic splines
- Stein's unbiased risk estimate
- Cauchy–Pompeiu formula
- Bump function
- Stein's lemma
- Padé approximant
- Smoothing spline
- Peano kernel theorem
- Stein's example
- Quasi-analytic function
- Quadratic convergence of Newton's method
- Analytic function between Banach spaces
- Blancmange curve
- Landau–Kolmogorov inequality
- Ramanujan summation
- Smoothness classes
- Delta method
- Regular surface
- Markov brothers' inequality
- Euler–Boole summation
- Wirtinger's inequality for functions
- Legendre transformation
- Wiman–Valiron theory
- Padé table
- Differentiable dependence on parameters
- Denjoy–Carleman theorem
- Borel's lemma
- Error bound for the trapezoidal rule
- Finite difference coefficients
- Heun's method
- Abel's summation formula
- Midpoint method
Further reading
- Web https://en.wikipedia.org/wiki/Differentiable_function, Differentiability of real functions of one variable
About this page
| Tag | 0A61B60B5R |
|---|---|
| URI | m2://0A61B60B5R |
| Kind | definition |
| Writing | W1 stated |
| MSC | 26A24 |
| Defines | Differentiable function |
| Uses | Derivative |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.