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Dini derivative

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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

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

is denoted by f′+ and defined by

f + ′ ( t ) = lim sup h → 0 + f ( t + h ) − f ( t ) h , {\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

f − ′ ( t ) = lim inf h → 0 + f ( t ) − f ( t − h ) h , {\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

f + ′ ( t , d ) = lim sup h → 0 + f ( t + h d ) − f ( t ) h . {\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).
  • Also,
D + f ( t ) = lim sup h → 0 + f ( t + h ) − f ( t ) h {\displaystyle D^{+}f(t)=\limsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}}}

and

D − f ( t ) = lim inf h → 0 + f ( t ) − f ( t − h ) h {\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
D + f ( t ) = lim inf h → 0 + f ( t + h ) − f ( t ) h {\displaystyle D_{+}f(t)=\liminf _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}}}

and

D − f ( t ) = lim sup h → 0 + f ( t ) − f ( t − h ) h {\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 ( D + f ( t ) = D + f ( t ) = D − f ( t ) = D − f ( t ) {\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

References

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

This article incorporates material from Dini derivative on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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