let T, S be non trivial RealNormSpace; :: thesis: for f being PartFunc of S,T
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) )
let f be PartFunc of S,T; :: thesis: for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) )
let x0 be Point of S; :: thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) ) by NFCONT_1:def 9; :: thesis: ( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) implies f is_continuous_in x0 )
assume A1: ( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) ; :: thesis: f is_continuous_in x0
hence x0 in dom f ; :: according to NFCONT_1:def 9 :: thesis: for b1 being Element of K21(K22(NAT ,the carrier of S)) holds
( not rng b1 c= dom f or not b1 is convergent or not lim b1 = x0 or ( f /* b1 is convergent & f /. x0 = lim (f /* b1) ) )
let s2 be sequence of S; :: thesis: ( not rng s2 c= dom f or not s2 is convergent or not lim s2 = x0 or ( f /* s2 is convergent & f /. x0 = lim (f /* s2) ) )
assume A2: ( rng s2 c= dom f & s2 is convergent & lim s2 = x0 ) ; :: thesis: ( f /* s2 is convergent & f /. x0 = lim (f /* s2) )
now
per cases ( ex n being Element of NAT st
for m being Element of NAT st n <= m holds
s2 . m = x0 or for n being Element of NAT ex m being Element of NAT st
( n <= m & s2 . m <> x0 ) ) ;
suppose ex n being Element of NAT st
for m being Element of NAT st n <= m holds
s2 . m = x0 ; :: thesis: ( f /* s2 is convergent & f /. x0 = lim (f /* s2) )
then consider N being Element of NAT such that
A3: for m being Element of NAT st N <= m holds
s2 . m = x0 ;
A4: for n being Element of NAT holds (s2 ^\ N) . n = x0
proof
let n be Element of NAT ; :: thesis: (s2 ^\ N) . n = x0
s2 . (n + N) = x0 by A3, NAT_1:12;
hence (s2 ^\ N) . n = x0 by NAT_1:def 3; :: thesis: verum
end;
A5: rng (s2 ^\ N) c= rng s2 by VALUED_0:21;
A6: now
let p be Real; :: thesis: ( p > 0 implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p )
assume A7: p > 0 ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p
take n = 0 ; :: thesis: for m being Element of NAT st n <= m holds
||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p
let m be Element of NAT ; :: thesis: ( n <= m implies ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p )
assume n <= m ; :: thesis: ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p
||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| = ||.((f /. ((s2 ^\ N) . m)) - (f /. x0)).|| by A2, A5, FUNCT_2:186, XBOOLE_1:1
.= ||.((f /. x0) - (f /. x0)).|| by A4
.= ||.(0. T).|| by RLVECT_1:28
.= 0 by NORMSP_1:5 ;
hence ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p by A7; :: thesis: verum
end;
then A8: f /* (s2 ^\ N) is convergent by NORMSP_1:def 9;
f /* (s2 ^\ N) = (f /* s2) ^\ N by A2, VALUED_0:27;
then ( f /* s2 is convergent & f /. x0 = lim ((f /* s2) ^\ N) ) by A6, A8, LOPBAN_3:15, NORMSP_1:def 11;
hence ( f /* s2 is convergent & f /. x0 = lim (f /* s2) ) by LOPBAN_3:14; :: thesis: verum
end;
suppose A9: for n being Element of NAT ex m being Element of NAT st
( n <= m & s2 . m <> x0 ) ; :: thesis: ( f /* s2 is convergent & f /. x0 = lim (f /* s2) )
then consider m1 being Element of NAT such that
A10: ( 0 <= m1 & s2 . m1 <> x0 ) ;
defpred S1[ Nat] means s2 . $1 <> x0;
A11: ex m being Nat st S1[m] by A10;
consider M being Nat such that
A12: ( S1[M] & ( for n being Nat st S1[n] holds
M <= n ) ) from NAT_1:sch 5(A11);
A13: now
let n be Element of NAT ; :: thesis: ex m being Element of NAT st
( n < m & s2 . m <> x0 )
consider m being Element of NAT such that
A14: ( n + 1 <= m & s2 . m <> x0 ) by A9;
take m = m; :: thesis: ( n < m & s2 . m <> x0 )
thus ( n < m & s2 . m <> x0 ) by A14, NAT_1:13; :: thesis: verum
end;
defpred S2[ Element of NAT , set , set ] means for n, m being Element of NAT st $2 = n & $3 = m holds
( n < m & s2 . m <> x0 & ( for k being Element of NAT st n < k & s2 . k <> x0 holds
m <= k ) );
A15: for n, x being Element of NAT ex y being Element of NAT st S2[n,x,y]
proof
let n, x be Element of NAT ; :: thesis: ex y being Element of NAT st S2[n,x,y]
defpred S3[ Nat] means ( x < $1 & s2 . $1 <> x0 );
ex m being Element of NAT st S3[m] by A13;
then A16: ex m being Nat st S3[m] ;
consider l being Nat such that
A17: ( S3[l] & ( for k being Nat st S3[k] holds
l <= k ) ) from NAT_1:sch 5(A16);
 reconsider l = l as Element of NAT by ORDINAL1:def 13;
take l ; :: thesis: S2[n,x,l]
thus S2[n,x,l] by A17; :: thesis: verum
end;
reconsider M' = M as Element of NAT by ORDINAL1:def 13;
consider F being Function of NAT ,NAT such that
A18: ( F . 0 = M' & ( for n being Element of NAT holds S2[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch 2(A15);
A19: ( dom F = NAT & rng F c= NAT ) by FUNCT_2:def 1;
rng F c= REAL by XBOOLE_1:1;
then reconsider F = F as Real_Sequence by A19, RELSET_1:11;
A20: now
let n be Element of NAT ; :: thesis: F . n is Element of NAT
F . n in rng F by A19, FUNCT_1:def 5;
hence F . n is Element of NAT by A19; :: thesis: verum
end;
now
let n be Element of NAT ; :: thesis: F . n < F . (n + 1)
( F . n is Element of NAT & F . (n + 1) is Element of NAT ) by A20;
hence F . n < F . (n + 1) by A18; :: thesis: verum
end;
then reconsider F = F as V35() sequence of NAT by SEQM_3:def 11;
A21: for n being Element of NAT st s2 . n <> x0 holds
ex m being Element of NAT st F . m = n
proof
defpred S3[ Nat] means ( s2 . $1 <> x0 & ( for m being Element of NAT holds F . m <> $1 ) );
assume ex n being Element of NAT st S3[n] ; :: thesis: contradiction
then A22: ex n being Nat st S3[n] ;
consider M1 being Nat such that
A23: ( S3[M1] & ( for n being Nat st S3[n] holds
M1 <= n ) ) from NAT_1:sch 5(A22);
 defpred S4[ Nat] means ( $1 < M1 & s2 . $1 <> x0 & ex m being Element of NAT st F . m = $1 );
A24: ex n being Nat st S4[n]
proof
take M ; :: thesis: S4[M]
A25: M <= M1 by A12, A23;
M <> M1 by A18, A23;
hence M < M1 by A25, XXREAL_0:1; :: thesis: ( s2 . M <> x0 & ex m being Element of NAT st F . m = M )
thus s2 . M <> x0 by A12; :: thesis: ex m being Element of NAT st F . m = M
take 0 ; :: thesis: F . 0 = M
thus F . 0 = M by A18; :: thesis: verum
end;
A26: for n being Nat st S4[n] holds
n <= M1 ;
consider MX being Nat such that
A27: ( S4[MX] & ( for n being Nat st S4[n] holds
n <= MX ) ) from NAT_1:sch 6(A26, A24);
 A28: for k being Element of NAT st MX < k & k < M1 holds
s2 . k = x0
proof
given k being Element of NAT such that A29: ( MX < k & k < M1 & s2 . k <> x0 ) ; :: thesis: contradiction
now
per cases ( ex m being Element of NAT st F . m = k or for m being Element of NAT holds F . m <> k ) ;
end;
end;
hence contradiction ; :: thesis: verum
end;
consider m being Element of NAT such that
A30: F . m = MX by A27;
A31: ( MX < F . (m + 1) & s2 . (F . (m + 1)) <> x0 & ( for k being Element of NAT st MX < k & s2 . k <> x0 holds
F . (m + 1) <= k ) ) by A18, A30;
M1 in NAT by ORDINAL1:def 13;
then A32: F . (m + 1) <= M1 by A18, A23, A27, A30;
now
assume F . (m + 1) <> M1 ; :: thesis: contradiction
then F . (m + 1) < M1 by A32, XXREAL_0:1;
hence contradiction by A28, A31; :: thesis: verum
end;
hence contradiction by A23; :: thesis: verum
end;
A33: for n being Element of NAT holds (s2 * F) . n <> x0
proof
defpred S3[ Element of NAT ] means (s2 * F) . $1 <> x0;
A34: S3[ 0 ] by A12, A18, FUNCT_2:21;
A35: for k being Element of NAT st S3[k] holds
S3[k + 1]
proof
let k be Element of NAT ; :: thesis: ( S3[k] implies S3[k + 1] )
assume (s2 * F) . k <> x0 ; :: thesis: S3[k + 1]
S2[k,F . k,F . (k + 1)] by A18;
then s2 . (F . (k + 1)) <> x0 ;
hence S3[k + 1] by FUNCT_2:21; :: thesis: verum
end;
thus for n being Element of NAT holds S3[n] from NAT_1:sch 1(A34, A35); :: thesis: verum
end;
A37: s2 * F is convergent by A2, LOPBAN_3:11;
A38: rng (s2 * F) c= rng s2 by VALUED_0:21;
then A39: rng (s2 * F) c= dom f by A2, XBOOLE_1:1;
lim (s2 * F) = x0 by A2, LOPBAN_3:12;
then A40: ( f /* (s2 * F) is convergent & f /. x0 = lim (f /* (s2 * F)) ) by A1, A33, A37, A39;
A41: now
let p be Real; :: thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s2) . m) - (f /. x0)).|| < p )
assume A42: 0 < p ; :: thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s2) . m) - (f /. x0)).|| < p
then consider n being Element of NAT such that
A43: for m being Element of NAT st n <= m holds
||.(((f /* (s2 * F)) . m) - (f /. x0)).|| < p by A40, NORMSP_1:def 11;
take k = F . n; :: thesis: for m being Element of NAT st k <= m holds
||.(((f /* s2) . m) - (f /. x0)).|| < p
let m be Element of NAT ; :: thesis: ( k <= m implies ||.(((f /* s2) . m) - (f /. x0)).|| < p )
assume A44: k <= m ; :: thesis: ||.(((f /* s2) . m) - (f /. x0)).|| < p
now
per cases ( s2 . m = x0 or s2 . m <> x0 ) ;
suppose A45: s2 . m = x0 ; :: thesis: ||.(((f /* s2) . m) - (f /. x0)).|| < p
||.(((f /* s2) . m) - (f /. x0)).|| = ||.((f /. (s2 . m)) - (f /. x0)).|| by A2, FUNCT_2:186
.= ||.(0. T).|| by A45, RLVECT_1:28
.= 0 by NORMSP_1:5 ;
hence ||.(((f /* s2) . m) - (f /. x0)).|| < p by A42; :: thesis: verum
end;
suppose s2 . m <> x0 ; :: thesis: ||.(((f /* s2) . m) - (f /. x0)).|| < p
then consider l being Element of NAT such that
A46: m = F . l by A21;
n <= l by A44, A46, SEQM_3:7;
then ||.(((f /* (s2 * F)) . l) - (f /. x0)).|| < p by A43;
then ||.((f /. ((s2 * F) . l)) - (f /. x0)).|| < p by A2, A38, FUNCT_2:186, XBOOLE_1:1;
then ||.((f /. (s2 . m)) - (f /. x0)).|| < p by A46, FUNCT_2:21;
hence ||.(((f /* s2) . m) - (f /. x0)).|| < p by A2, FUNCT_2:186; :: thesis: verum
end;
end;
end;
hence ||.(((f /* s2) . m) - (f /. x0)).|| < p ; :: thesis: verum
end;
hence f /* s2 is convergent by NORMSP_1:def 9; :: thesis: f /. x0 = lim (f /* s2)
hence f /. x0 = lim (f /* s2) by A41, NORMSP_1:def 11; :: thesis: verum
end;
end;
end;
hence ( f /* s2 is convergent & f /. x0 = lim (f /* s2) ) ; :: thesis: verum