Dini derivative

In the analysis of functions of real variables , the Dini derivatives are one of the generalizations of the concept of a derivative .

Upper Dini Derived Continuous Function

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

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

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

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

f'_{+}(t)\triangleq \varlimsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}}

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

f '_ {+} (t) \ triangleq \ varlimsup _ {{h \ to {0 +}}} {\ frac {f (t + h) -f (t)} {h}}

,

Where $\varlimsup$ ${\ displaystyle \ varlimsup}$ $\ varlimsup$ there is an upper partial limit .

Dini lower derivative , $f'_{-},$ ${\ displaystyle f '_ {-},}$ ${\ displaystyle f '_ {-},}$ defined as

f'_{-}(t)\triangleq \varliminf _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}}

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

f '_ {-} (t) \ triangleq \ varliminf _ {{h \ to {0 +}}} {\ frac {f (t + h) -f (t)} {h}}

,

Where $\varliminf$ ${\ displaystyle \ varliminf}$ $\ varliminf$ there is a lower partial limit .

If a $f$ ${\ displaystyle f}$ $f$ is defined on the vector space , then the upper Dini derivative at the point $t$ ${\ displaystyle t}$ $t$ towards $d$ ${\ displaystyle d}$ $d$ defined as

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

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

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

If a $f$ ${\ displaystyle f}$ $f$ locally Lipschitz (i.e., each point has a neighborhood , the restriction $f$ ${\ displaystyle f}$ $f$ which is the Lipschitz function), then $f'_{+}$ ${\ displaystyle f '_ {+}}$ ${\ displaystyle f '_ {+}}$ finite. If a $f$ ${\ displaystyle f}$ $f$ differentiable at a point $t$ ${\ displaystyle t}$ $t$ , then the Dini derivative at this point coincides with the ordinary derivative in $t$ ${\ displaystyle t}$ $t$ .

Notes

Sometimes use the notation $D^{+}f(t)$ ${\ displaystyle D ^ {+} f (t)}$ instead $f'_{+}(t),$ ${\ displaystyle f '_ {+} (t),}$ and $D_{+}f(t)$ ${\ displaystyle D _ {+} f (t)}$ used instead $f'_{-}(t).$ ${\ displaystyle f '_ {-} (t).}$

Also use the notation

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

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

and

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

{\ displaystyle D _ {-} f (t) \ triangleq \ varliminf _ {h \ to {0 -}} {\ frac {f (t + h) -f (t)} {h}}}

So when used $D$ ${\ displaystyle D}$ Annotation of derivatives Dini, plus and minus signs indicate the left-side or right-hand limit , and the position of the sign indicates the type of derivative (upper or lower).

On the extended number line, each of the Dini derivatives always exists, however, they can sometimes take values $+\infty$ ${\ displaystyle + \ infty}$ or $-\infty$ ${\ displaystyle - \ infty}$

Literature

Royden, HL Real analysis. - 2nd. - MacMillan, 1968. - ISBN 978-0-02-404150-0 .

Dini derivative

Notes

Literature

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