Dini derivative explained

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},

is denoted by and defined by

f' _{+(t)} = \limsup _{h} } \frac,

where is the supremum limit and the limit is a one-sided limit. The lower Dini derivative,, is defined by

f' _{-(t)} = \liminf _{h} } \frac,

where is the infimum limit.

If is defined on a vector space, then the upper Dini derivative at in the direction is defined by

f' _{+} (t,d) = \limsup _{h} } \frac.

If is locally Lipschitz, then is finite. If is differentiable at, then the Dini derivative at is the usual derivative at .

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 on the real line, only if all the Dini derivatives exist, and have the same value.
• Sometimes the notation is used instead of and is used instead of .^[1]
• Also,

D ^{+} f(t) = \limsup _{h} } \frac

and

D _{-} f(t) = \liminf _{h} } \frac.

• So when using the 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) = \liminf _{h} } \frac

and

D ^{-} f(t) = \limsup _{h} } \frac.

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) ) then the function is differentiable in the usual sense at the point  .

• 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

• .
• Book: Royden, H. L. . Real Analysis . MacMillan . 1968 . 2nd . 978-0-02-404150-0.
• Book: Brian S. . Thomson. Judith B. . Bruckner. Andrew M. . Bruckner. Elementary Real Analysis. 2008. ClassicalRealAnalysis.com [first edition published by Prentice Hall in 2001]. 978-1-4348-4161-2. 301–302.

Notes and References

1. Book: Khalil, Hassan K. . 2002 . 3rd . Nonlinear Systems . 0-13-067389-7 . . Upper Saddle River, NJ.