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

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).^[1]
• 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

• Denjoy–Young–Saks theorem – Mathematical theorem about Dini derivatives
• Derivative (generalizations) – Fundamental construction of differential calculusPages displaying short descriptions of redirect targets
• Semi-differentiability – Property of a mathematical function

References

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