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, 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).

Dini derivative

Remarks

See also