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In the mathematics , and specifically real analysis , the Dini derivatives (or Dini derivates ) are a class of generalizations of the derivative . The upper Dini derivative of a continuous function ,
f
:
R
→
R
,
{\displaystyle f:{\mathbb {R} }\rightarrow {\mathbb {R} },}
denoted by
f
+
′
,
{\displaystyle f'_{+},\,}
f
+
′
(
t
)
=
lim
h
→
0
+
sup
f
(
t
+
h
)
−
f
(
t
)
h
.
{\displaystyle f'_{+}(t)=\lim _{h\rightarrow 0^{+}}\sup {\frac {f(t+h)-f(t)}{h}}.}
The lower Dini derivative ,
f
−
′
,
{\displaystyle f'_{-},\,}
f
−
′
(
t
)
=
lim
h
→
0
+
inf
f
(
t
+
h
)
−
f
(
t
)
h
{\displaystyle f'_{-}(t)=\lim _{h\rightarrow 0^{+}}\inf {\frac {f(t+h)-f(t)}{h}}}
(see lim sup and lim inf ). If
f
{\displaystyle f}
vector space , then the upper Dini derivative at
t
{\displaystyle t}
d
{\displaystyle d}
f
+
′
(
t
,
d
)
=
lim
h
→
0
+
sup
f
(
t
+
h
d
)
−
f
(
t
)
h
.
{\displaystyle f'_{+}(t,d)=\lim _{h\rightarrow 0^{+}}\sup {\frac {f(t+hd)-f(t)}{h}}.}
If
f
{\displaystyle f}
locally Lipschitz then
f
+
′
{\displaystyle f'_{+}\,}
f
{\displaystyle f}
differentiable at
t
{\displaystyle t}
t
{\displaystyle t}
derivative at
t
.
{\displaystyle t.}
Sometimes the notation
D
+
f
(
t
)
{\displaystyle D^{+}f(t)\,}
f
+
′
(
t
)
,
{\displaystyle f'_{+}(t),\,}
D
+
f
(
t
)
{\displaystyle D_{+}f(t)\,}
f
−
′
(
t
)
.
{\displaystyle f'_{-}(t).\,}
D
−
f
(
t
)
=
lim
h
→
0
−
sup
f
(
t
+
h
)
−
f
(
t
)
h
.
{\displaystyle D^{-}f(t)=\lim _{h\rightarrow 0^{-}}\sup {\frac {f(t+h)-f(t)}{h}}.}
and
D
−
f
(
t
)
=
lim
h
→
0
−
inf
f
(
t
+
h
)
−
f
(
t
)
h
.
{\displaystyle D_{-}f(t)=\lim _{h\rightarrow 0^{-}}\inf {\frac {f(t+h)-f(t)}{h}}.}
So when using the
D
{\displaystyle D}
l
i
m
i
n
f
{\displaystyle liminf}
l
i
m
s
u
p
{\displaystyle limsup}
Each of the Dini derivatives always exist; however, they may take on the values
+
∞
{\displaystyle +\infty }
−
∞
{\displaystyle -\infty }
See also
References
Dini derivative at PlanetMath .
