Dini derivative - Alchetron, The Free Social Encyclopedia

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.

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

f : R → R ,

is denoted by f′+ and defined by

f + ′ ( t ) ≜ lim sup h → 0 + 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 + h ) − f ( t ) 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 .

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

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 ) − f ( t − h ) h

and

D − f ( t ) ≜ lim inf h → 0 + 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.

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).

References

Dini derivative Wikipedia

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