Peano derivative
One of the generalizations of the concept of a derivative. Let there exist a 
such that for all 
with 
one has
 |
where 
are constants and 
as 
; let 
. Then 
is called the generalized Peano derivative of order 
of the function 
at the point 
. Symbol: 
; in particular, 
, 
. If 
exists, then 
, 
, also exists. If the finite ordinary two-sided derivative 
exists, then 
. The converse is false for 
: For the function
 |
one has 
, 
but 
does not exist for 
(since 
is discontinuous for 
). Consequently, the ordinary derivative 
does not exist for 
.
Infinite generalized Peano derivatives have also been introduced. Let for all 
with 
,
 |
where 
are constants and 
as 
(
is a number or the symbol 
). Then 
is also called the Peano derivative of order 
of the function 
at the point 
. It was introduced by G. Peano.
Peano derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peano_derivative&oldid=48146