let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom f & ex r being Real st rng f = {r} holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) )
let f be PartFunc of REAL,REAL; ( Z c= dom f & ex r being Real st rng f = {r} implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) ) )
reconsider cf = REAL --> (In (0,REAL)) as Function of REAL,REAL ;
set R = cf;
then reconsider R = cf as RestFunc by Def2;
set L = cf;
for p being Real holds cf . p = 0 * p
;
then reconsider L = cf as LinearFunc by Def3;
assume A7:
Z c= dom f
; ( for r being Real holds not rng f = {r} or ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) ) )
given r being Real such that A8:
rng f = {r}
; ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) )
hence A14:
f is_differentiable_on Z
by A7, Th9; for x being Real st x in Z holds
(f `| Z) . x = 0
let x0 be Real; ( x0 in Z implies (f `| Z) . x0 = 0 )
assume A15:
x0 in Z
; (f `| Z) . x0 = 0
then A16:
f is_differentiable_in x0
by A10;
then
ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )
;
then consider N being Neighbourhood of x0 such that
A17:
N c= dom f
;
A18:
for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))
reconsider j = 1 as Element of REAL by XREAL_0:def 1;
thus (f `| Z) . x0 =
diff (f,x0)
by A14, A15, Def7
.=
L . j
by A16, A17, A18, Def5
.=
0
; verum