We
continue
We use the previous result
:
and conclude
.
We continue similarly for any
:
for any derivative
there will be a term
and the rest of the terms are zero at
by results of the previous steps.
Proof
(b)
(c)
Set
By condition (1) the
is defined for all
except, possibly,
.
We
have
Hence, if (b) takes place
then
We do Taylor decomposition, (see the proposition
(
Taylor
decomposition in Schlomilch, Lagrange and Cauchy
forms
)):
Thus, by (b) and
,
The
then must be of the
form
for a finite sequence
.
This is (c).
The calculation may be performed in opposite direction: assume (c), then the
has the
form
and we do argument in reverse arriving to (b).
Proof
(b)
(d)
The proof is a direct verification. We use finiteness of
to
differentiate
termwise and substitute into (b).
