A1: {} REAL is closed by XBOOLE_1:3;
( ([#] REAL) ` = {} REAL & REAL c= REAL & [#] REAL = REAL ) by XBOOLE_1:37;
then reconsider Z = [#] REAL as open Subset of REAL by ;
reconsider cf = REAL --> (In (0,REAL)) as Function of REAL,REAL ;
set R = cf;
reconsider f = cf as PartFunc of REAL,REAL ;
now :: thesis: for h being non-zero 0 -convergent Real_Sequence holds
( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0 )
let h be non-zero 0 -convergent Real_Sequence; :: thesis: ( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0 )
A3: now :: thesis: for n being Nat holds ((h ") (#) (cf /* h)) . n = 0
let n be Nat; :: thesis: ((h ") (#) (cf /* h)) . n = 0
A4: rng h c= dom cf ;
A6: n in NAT by ORDINAL1:def 12;
thus ((h ") (#) (cf /* h)) . n = ((h ") . n) * ((cf /* h) . n) by SEQ_1:8
.= ((h ") . n) * (cf . (h . n)) by
.= ((h ") . n) * 0
.= 0 ; :: thesis: verum
end;
then A7: (h ") (#) (cf /* h) is V8() by VALUED_0:def 18;
hence (h ") (#) (cf /* h) is convergent ; :: thesis: lim ((h ") (#) (cf /* h)) = 0
((h ") (#) (cf /* h)) . 0 = 0 by A3;
hence lim ((h ") (#) (cf /* h)) = 0 by ; :: thesis: verum
end;
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;
f | Z is V8() ;
then consider r being Element of REAL such that
A8: for x being Element of REAL st x in Z /\ (dom f) holds
f . x = r by PARTFUN2:57;
A9: now :: thesis: for x0 being Real st x0 in Z holds
f is_differentiable_in x0
let x0 be Real; :: thesis: ( x0 in Z implies f is_differentiable_in x0 )
reconsider xx0 = x0 as Element of REAL by XREAL_0:def 1;
assume x0 in Z ; :: thesis:
set N = the Neighbourhood of x0;
A11: xx0 in Z /\ (dom f) ;
for x being Real st x in the Neighbourhood of x0 holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))
proof
let x be Real; :: thesis: ( x in the Neighbourhood of x0 implies (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )
reconsider xx = x as Element of REAL by XREAL_0:def 1;
assume x in the Neighbourhood of x0 ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))
then x in Z /\ (dom f) ;
hence (f . x) - (f . x0) = r - (f . x0) by A8
.= r - r by
.= (L . (xx - xx0)) + 0
.= (L . (x - x0)) + (R . (x - x0)) ;
:: thesis: verum
end;
hence f is_differentiable_in x0 ; :: thesis: verum
end;
set x0 = the Element of REAL ;
f is_differentiable_in the Element of REAL by A9;
then consider N being Neighbourhood of the Element of REAL such that
N c= dom f and
A12: ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
(f . x) - (f . the Element of REAL ) = (L . (x - the Element of REAL )) + (R . (x - the Element of REAL )) ;
consider L being LinearFunc, R being RestFunc such that
A13: for x being Real st x in N holds
(f . x) - (f . the Element of REAL ) = (L . (x - the Element of REAL )) + (R . (x - the Element of REAL )) by A12;
take R ; :: thesis:
consider p being Real such that
A14: for r being Real holds L . r = p * r by Def3;
(f . the Element of REAL ) - (f . the Element of REAL ) = (L . ( the Element of REAL - the Element of REAL )) + (R . ( the Element of REAL - the Element of REAL )) by ;
then A15: 0 = (p * 0) + (R . 0) by A14;
hence R . 0 = 0 ; :: thesis:
A16: now :: thesis: for h being non-zero 0 -convergent Real_Sequence holds
( R /* h is convergent & lim (R /* h) = R . 0 )
set s3 = seq_const 0;
let h be non-zero 0 -convergent Real_Sequence; :: thesis: ( R /* h is convergent & lim (R /* h) = R . 0 )
A17: (seq_const 0) . 1 = 0 ;
(h ") (#) (R /* h) is convergent by Def2;
then (h ") (#) (R /* h) is bounded ;
then consider M being Real such that
M > 0 and
A18: for n being Nat holds |.(((h ") (#) (R /* h)) . n).| < M by SEQ_2:3;
A19: now :: thesis: for n being Nat holds
( () . n <= (abs (R /* h)) . n & (abs (R /* h)) . n <= (M (#) (abs h)) . n )
let n be Nat; :: thesis: ( () . n <= (abs (R /* h)) . n & (abs (R /* h)) . n <= (M (#) (abs h)) . n )
|.(((h ") (#) (R /* h)) . n).| = |.(((h ") . n) * ((R /* h) . n)).| by SEQ_1:8
.= |.(((h . n) ") * ((R /* h) . n)).| by VALUED_1:10 ;
then A20: |.(((h . n) ") * ((R /* h) . n)).| <= M by A18;
0 <= |.((R /* h) . n).| by COMPLEX1:46;
then 0 <= (abs (R /* h)) . n by SEQ_1:12;
hence (seq_const 0) . n <= (abs (R /* h)) . n ; :: thesis: (abs (R /* h)) . n <= (M (#) (abs h)) . n
|.(h . n).| >= 0 by COMPLEX1:46;
then |.(h . n).| * |.(((h . n) ") * ((R /* h) . n)).| <= M * |.(h . n).| by ;
then |.((h . n) * (((h . n) ") * ((R /* h) . n))).| <= M * |.(h . n).| by COMPLEX1:65;
then A21: |.(((h . n) * ((h . n) ")) * ((R /* h) . n)).| <= M * |.(h . n).| ;
h . n <> 0 by SEQ_1:5;
then |.(1 * ((R /* h) . n)).| <= M * |.(h . n).| by ;
then |.((R /* h) . n).| <= M * ((abs h) . n) by SEQ_1:12;
then |.((R /* h) . n).| <= (M (#) (abs h)) . n by SEQ_1:9;
hence (abs (R /* h)) . n <= (M (#) (abs h)) . n by SEQ_1:12; :: thesis: verum
end;
lim h = 0 ;
then lim (abs h) = by SEQ_4:14
.= 0 by ABSVALUE:2 ;
then A22: lim (M (#) (abs h)) = M * 0 by SEQ_2:8
.= lim () by ;
A23: abs (R /* h) is convergent by ;
lim () = 0 by ;
then lim (abs (R /* h)) = 0 by ;
hence ( R /* h is convergent & lim (R /* h) = R . 0 ) by ; :: thesis: verum
end;
now :: thesis: for s1 being Real_Sequence st rng s1 c= dom R & s1 is convergent & lim s1 = 0 & ( for n being Nat holds s1 . n <> 0 ) holds
( R /* s1 is convergent & lim (R /* s1) = R . 0 )
let s1 be Real_Sequence; :: thesis: ( rng s1 c= dom R & s1 is convergent & lim s1 = 0 & ( for n being Nat holds s1 . n <> 0 ) implies ( R /* s1 is convergent & lim (R /* s1) = R . 0 ) )
assume that
rng s1 c= dom R and
A24: ( s1 is convergent & lim s1 = 0 ) and
A25: for n being Nat holds s1 . n <> 0 ; :: thesis: ( R /* s1 is convergent & lim (R /* s1) = R . 0 )
s1 is 0 -convergent by A24;
then s1 is non-zero 0 -convergent Real_Sequence by ;
hence ( R /* s1 is convergent & lim (R /* s1) = R . 0 ) by A16; :: thesis: verum
end;
hence R is_continuous_in 0 by FCONT_1:2; :: thesis: verum