per cases ( f | X is empty or not f | X is empty ) ;

suppose f | X is empty ; :: thesis: for b₁ being PartFunc of REAL,REAL st b₁ = f | X holds
b₁ is continuous

hence for b₁ being PartFunc of REAL,REAL st b₁ = f | X holds
b₁ is continuous ; :: thesis: verum

end;

suppose not f | X is empty ; :: thesis: for b₁ being PartFunc of REAL,REAL st b₁ = f | X holds
b₁ is continuous

then consider x0 being Real such that
A1: x0 in dom (f | X) by MEMBERED:9;
x0 in X by A1, RELAT_1:57;
then A2: X = {x0} by ZFMISC_1:132;

now :: thesis: for p being Real st p in dom (f | X) holds
f | X is_continuous_in p

let p be Real; :: thesis: ( p in dom (f | X) implies f | X is_continuous_in p )
assume p in dom (f | X) ; :: thesis: f | X is_continuous_in p
then A3: p in {x0} by A2;
thus f | X is_continuous_in p :: thesis: verum

proof

let s1 be Real_Sequence; :: according to FCONT_1:def 1 :: thesis: ( rng s1 c= dom (f | X) & s1 is convergent & lim s1 = p implies ( (f | X) /* s1 is convergent & (f | X) . p = lim ((f | X) /* s1) ) )
assume that
A4: rng s1 c= dom (f | X) and
s1 is convergent and
lim s1 = p ; :: thesis: ( (f | X) /* s1 is convergent & (f | X) . p = lim ((f | X) /* s1) )
A5: (dom f) /\ {x0} c= {x0} by XBOOLE_1:17;
rng s1 c= (dom f) /\ {x0} by A2, A4, RELAT_1:61;
then A6: rng s1 c= {x0} by A5;

A7: now :: thesis: for n being Element of NAT holds s1 . n = x0

let n be Element of NAT ; :: thesis: s1 . n = x0
s1 . n in rng s1 by VALUED_0:28;
hence s1 . n = x0 by A6, TARSKI:def 1; :: thesis: verum

end;

A8: p = x0 by A3, TARSKI:def 1;

A9: now :: thesis: for g being Real st 0 < g holds
ex n being Nat st
for m being Nat st n <= m holds
|.((((f | {x0}) /* s1) . m) - ((f | {x0}) . p)).| < g

let g be Real; :: thesis: ( 0 < g implies ex n being Nat st
for m being Nat st n <= m holds
|.((((f | {x0}) /* s1) . m) - ((f | {x0}) . p)).| < g )
assume A10: 0 < g ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.((((f | {x0}) /* s1) . m) - ((f | {x0}) . p)).| < g
reconsider n = 0 as Nat ;
take n = n; :: thesis: for m being Nat st n <= m holds
|.((((f | {x0}) /* s1) . m) - ((f | {x0}) . p)).| < g
let m be Nat; :: thesis: ( n <= m implies |.((((f | {x0}) /* s1) . m) - ((f | {x0}) . p)).| < g )
assume n <= m ; :: thesis: |.((((f | {x0}) /* s1) . m) - ((f | {x0}) . p)).| < g
A11: m in NAT by ORDINAL1:def 12;
|.((((f | {x0}) /* s1) . m) - ((f | {x0}) . p)).| = |.(((f | {x0}) . (s1 . m)) - ((f | {x0}) . x0)).| by A2, A8, A4, FUNCT_2:108, A11
.= |.(((f | {x0}) . x0) - ((f | {x0}) . x0)).| by A7, A11
.= 0 by ABSVALUE:2 ;
hence |.((((f | {x0}) /* s1) . m) - ((f | {x0}) . p)).| < g by A10; :: thesis: verum

end;

hence (f | X) /* s1 is convergent by A2, SEQ_2:def 6; :: thesis: (f | X) . p = lim ((f | X) /* s1)
hence (f | X) . p = lim ((f | X) /* s1) by A2, A9, SEQ_2:def 7; :: thesis: verum

end;

end;

hence for b₁ being PartFunc of REAL,REAL st b₁ = f | X holds
b₁ is continuous ; :: thesis: verum

end;

end;