Dini 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,^[1] of a continuous function

${\displaystyle f:{\mathbb {R} }\rightarrow {\mathbb {R} },}$

is denoted by f′+ and defined by

${\displaystyle f'_{+}(t)=\limsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}},}$

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

${\displaystyle f'_{-}(t)=\liminf _{h\to {0+}}{\frac {f(t)-f(t-h)}{h}},}$

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

${\displaystyle f'_{+}(t,d)=\limsup _{h\to {0+}}{\frac {f(t+hd)-f(t)}{h}}.}$

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).^[1]
• Also,

${\displaystyle D^{+}f(t)=\limsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}}}$ 

and

${\displaystyle D_{-}f(t)=\liminf _{h\to {0+}}{\frac {f(t)-f(t-h)}{h}}}$ .

• 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

${\displaystyle D_{+}f(t)=\liminf _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}}}$ 

and

${\displaystyle D^{-}f(t)=\limsup _{h\to {0+}}{\frac {f(t)-f(t-h)}{h}}}$ .

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 (${\displaystyle D^{+}f(t)=D_{+}f(t)=D^{-}f(t)=D_{-}f(t)}$ ) 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
• Derivative (generalizations)
• Semi-differentiability

References

1. Khalil, Hassan K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7.

• Lukashenko, T.P. (2001) [1994], "Dini derivative", Encyclopedia of Mathematics, EMS Press.
• 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.

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