Dini derivative

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In mathematics, the upper Dini derivative of a continuous function,

f:{\mathbb R} \mapsto {\mathbb R},

denoted by f'_+,\, is defined as

f'_+(t) = \lim_{h\rightarrow 0^+} \sup \frac{f(t + h) - f(t)}{h}.

The lower Dini derivative, f'_-,\, is defined as

f'_-(t) = \lim_{h\rightarrow 0^+} \inf \frac{f(t + h) - f(t)}{h}

(see lim sup and lim inf). If f is defined on a vector space, then the upper Dini derivative at t in the direction d is denoted

f'_+ (t,d) = \lim_{h\rightarrow 0^+} \sup \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.

[edit] Remarks

  • Sometimes the notation D^+ f(t)\, is used instead of f'_+(t),\, and D^-f(t)\, is used instead of f'_-(t).\,
  • Like conventional derivatives, Dini derivatives do not always exist.

[edit] See also

This article incorporates material from Dini derivative on PlanetMath, which is licensed under the GFDL.

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