Dini derivative

E877691

Dini derivative Dini derivative generalized derivative mathematical concept

The Dini derivative is a generalized notion of derivative that captures one-sided limiting rates of change of a function, even at points where the classical derivative may not exist.

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Observed surface forms (4)

Surface form	Occurrences
upper right Dini derivative	0
lower left Dini derivative	0
lower right Dini derivative	0
upper left Dini derivative	0

Statements (47)

Predicate	Object
instanceOf	Dini derivative ⓘ Dini derivative ⓘ Dini derivative ⓘ Dini derivative ⓘ generalized derivative ⓘ mathematical concept ⓘ
alsoCalled	upper right-hand Dini derivative ⓘ
appearsIn	criteria for absolute continuity on intervals ⓘ results on differentiability of convex functions ⓘ results on differentiability of monotone functions ⓘ
appliesTo	functions of a real variable ⓘ real-valued functions ⓘ
canBe	infinite ⓘ undefined at some points ⓘ
captures	one-sided limiting rates of change ⓘ
characterizes	monotone functions via nonnegative one-sided Dini derivatives almost everywhere ⓘ
comparedWith	Clarke generalized derivative NERFINISHED ⓘ subderivative ⓘ
definedAs	liminf_{h→0+} (f(x+h)−f(x))/h ⓘ liminf_{h→0−} (f(x+h)−f(x))/h ⓘ limsup_{h→0+} (f(x+h)−f(x))/h ⓘ limsup_{h→0−} (f(x+h)−f(x))/h ⓘ
domain	points of the real line ⓘ
field	mathematical analysis ⓘ real analysis ⓘ
generalizes	classical derivative ⓘ
hasProperty	one-sided nature ⓘ uses limsup or liminf instead of limit ⓘ
hasVariant	lower left Dini derivative ⓘ lower right Dini derivative ⓘ upper left Dini derivative ⓘ upper right Dini derivative ⓘ
implies	classical derivative exists when all four Dini derivatives coincide and are finite ⓘ
namedAfter	Ulisse Dini NERFINISHED ⓘ
range	extended real numbers ⓘ
relatedTo	liminf ⓘ limsup ⓘ
usedFor	describing local behavior of functions ⓘ studying absolute continuity ⓘ studying differentiability properties ⓘ studying monotonicity ⓘ
usedIn	Lyapunov stability theory NERFINISHED ⓘ control theory ⓘ differential inclusions ⓘ nonsmooth analysis ⓘ theory of functions of a real variable ⓘ
usedWhen	classical derivative does not exist ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Ulisse Dini → notableConcept → Dini derivative ⓘ