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.

The upper Dini derivative, which is also called an upper right-hand derivative,^[1] of a continuous function

$f:{{\mathbb R}}\rightarrow {{\mathbb R}},$

is denoted by $f'_{+},\,$ and defined by

${\displaystyle f'_{+}(t)\triangleq \limsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}},}$

where $\limsup$ is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, $f'_{-},\,$, is defined by

${\displaystyle f'_{-}(t)\triangleq \liminf _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}},}$

where $\liminf$ 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

$f'_{+}(t,d)\triangleq \limsup _{{h\to {0+}}}{\frac {f(t+hd)-f(t)}{h}}.$

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

• Sometimes the notation $D^{+}f(t)\,$ is used instead of $f'_{+}(t),\,$ and $D_{+}f(t)\,$ is used instead of $f'_{-}(t).\,$^[1]
• Also,

$D^{-}f(t)\triangleq \limsup _{{h\to {0-}}}{\frac {f(t+h)-f(t)}{h}}$

and

$D_{-}f(t)\triangleq \liminf _{{h\to {0-}}}{\frac {f(t+h)-f(t)}{h}}.$

• 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 $+\infty$ or $-\infty$ at times (i.e., the Dini derivatives always exist in the extended sense).

See also

• Denjoy–Young–Saks theorem
• Derivative (generalizations)

References

In-line references

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

General references

• Lukashenko, T.P. (2001), "Dini derivative", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
• Royden, H.L. (1968). Real analysis (2nd ed.). MacMillan. ISBN 978-0-02-404150-0.
• Brian S. Thomson; Judith B. Bruckner; Andrew M. Bruckner (2008). Elementary Real Analysis. ClassicalRealAnalysis.com [first edition published by Prentice Hall in 2001]. pp. 301–302. ISBN 978-1-4348-4161-2.

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