Dini derivative

Short description: Class of generalisations of the derivative

In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions.

The upper Dini derivative, which is also called an upper right-hand derivative, of a continuous function

[math]\displaystyle{ f:{\mathbb R} \rightarrow {\mathbb R}, }[/math]

is denoted by f′+ and defined by

[math]\displaystyle{ f'_+(t) = \limsup_{h \to {0+}} \frac{f(t + h) - f(t)}{h}, }[/math]

where lim sup is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, f′−, is defined by

[math]\displaystyle{ f'_-(t) = \liminf_{h \to {0+}} \frac{f(t) - f(t - h)}{h}, }[/math]

where lim inf is the infimum limit.

If f is defined on a vector space, then the upper Dini derivative at t in the direction d is defined by

[math]\displaystyle{ f'_+ (t,d) = \limsup_{h \to {0+}} \frac{f(t + hd) - f(t)}{h}. }[/math]

If f is locally Lipschitz, then f′+ is finite. If f is differentiable at t, then the Dini derivative at t is the usual derivative at t.

Remarks

• The functions are defined in terms of the infimum and supremum in order to make the Dini derivatives as "bullet proof" as possible, so that the Dini derivatives are well-defined for almost all functions, even for functions that are not conventionally differentiable. The upshot of Dini's analysis is that a function is differentiable at the point t on the real line (ℝ), only if all the Dini derivatives exist, and have the same value.

• Sometimes the notation D⁺ f(t) is used instead of f′+(t) and D₋ f(t) is used instead of f′−(t).
• Also,

[math]\displaystyle{ D^{+}f(t) = \limsup_{h \to {0+}} \frac{f(t + h) - f(t)}{h} }[/math]

and

[math]\displaystyle{ D_{-}f(t) = \liminf_{h \to {0+}} \frac{f(t) - f(t - h)}{h} }[/math].

• So when using the D notation of the Dini derivatives, the plus or minus sign indicates the left- or right-hand limit, and the placement of the sign indicates the infimum or supremum limit.

• There are two further Dini derivatives, defined to be

[math]\displaystyle{ D_{+}f(t) = \liminf_{h \to {0+}} \frac{f(t + h) - f(t)}{h} }[/math]

and

[math]\displaystyle{ D^{-}f(t) = \limsup_{h \to {0+}} \frac{f(t) - f(t - h)}{h} }[/math].

which are the same as the first pair, but with the supremum and the infimum reversed. For only moderately ill-behaved functions, the two extra Dini derivatives aren't needed. For particularly badly behaved functions, if all four Dini derivatives have the same value ([math]\displaystyle{ D^{+}f(t) = D_{+}f(t) = D^{-}f(t) = D_{-}f(t) }[/math]) then the function f is differentiable in the usual sense at the point t .

• On the extended reals, each of the Dini derivatives always exist; however, they may take on the values +∞ or −∞ at times (i.e., the Dini derivatives always exist in the extended sense).

See also

• Denjoy–Young–Saks theorem – Mathematical theorem about Dini derivatives
• Semi-differentiability

References

• Hazewinkel, Michiel, ed. (2001), "Dini derivative", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=d/d032530 .
• Royden, H. L. (1968). Real Analysis (2nd ed.). MacMillan. ISBN 978-0-02-404150-0. 
• Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008). Elementary Real Analysis. ClassicalRealAnalysis.com [first edition published by Prentice Hall in 2001]. pp. 301–302. ISBN 978-1-4348-4161-2.