let X be set ; :: thesis: for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st X c= dom f holds
( f | X is continuous iff for x0 being real number
for r being Real st x0 in X & 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )
let S be RealNormSpace; :: thesis: for f being PartFunc of REAL, the carrier of S st X c= dom f holds
( f | X is continuous iff for x0 being real number
for r being Real st x0 in X & 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )
let f be PartFunc of REAL, the carrier of S; :: thesis: ( X c= dom f implies ( f | X is continuous iff for x0 being real number
for r being Real st x0 in X & 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) )
assume A1: X c= dom f ; :: thesis: ( f | X is continuous iff for x0 being real number
for r being Real st x0 in X & 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )
thus ( f | X is continuous implies for x0 being real number
for r being Real st x0 in X & 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) :: thesis: ( ( for x0 being real number
for r being Real st x0 in X & 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f | X is continuous )

proof

assume A2: f | X is continuous ; :: thesis: for x0 being real number
for r being Real st x0 in X & 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
let x0 be real number ; :: thesis: for r being Real st x0 in X & 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
let r be Real; :: thesis: ( x0 in X & 0 < r implies ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; :: thesis: ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
x0 in dom (f | X) by A1, A3, RELAT_1:62;
then f | X is_continuous_in x0 by A2, Def2;
then consider s being real number such that
A5: 0 < s and
A6: for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A4, Th8;
take s ; :: thesis: ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
thus 0 < s by A5; :: thesis: for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r
let x1 be real number ; :: thesis: ( x1 in X & abs (x1 - x0) < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A7: x1 in X and
A8: abs (x1 - x0) < s ; :: thesis: ||.((f /. x1) - (f /. x0)).|| < r
B0: x0 in REAL by XREAL_0:def 1;
B1: x1 in REAL by XREAL_0:def 1;
A9: dom (f | X) = (dom f) /\ X by RELAT_1:61
.= X by A1, XBOOLE_1:28 ;
then ||.((f /. x1) - (f /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A7, PARTFUN2:15, B1
.= ||.(((f | X) /. x1) - ((f | X) /. x0)).|| by A3, A9, PARTFUN2:15, B0 ;
hence ||.((f /. x1) - (f /. x0)).|| < r by A6, A9, A7, A8; :: thesis: verum

end;

assume A10: for x0 being real number
for r being Real st x0 in X & 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ; :: thesis: f | X is continuous
A11: dom (f | X) = (dom f) /\ X by RELAT_1:61
.= X by A1, XBOOLE_1:28 ;

now

let x0 be real number ; :: thesis: ( x0 in dom (f | X) implies f | X is_continuous_in x0 )
assume A12: x0 in dom (f | X) ; :: thesis: f | X is_continuous_in x0
then A13: x0 in X by RELAT_1:57;
for r being Real st 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )

proof

let r be Real; :: thesis: ( 0 < r implies ex s being real number st
( 0 < s & ( for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) ) )
assume 0 < r ; :: thesis: ex s being real number st
( 0 < s & ( for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
then consider s being real number such that
A14: 0 < s and
A15: for x1 being real number st x1 in X & abs (x1 - x0) < s holds
||.((f /. x1) - (f /. x0)).|| < r by A10, A13;
take s ; :: thesis: ( 0 < s & ( for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
thus 0 < s by A14; :: thesis: for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
let x1 be real number ; :: thesis: ( x1 in dom (f | X) & abs (x1 - x0) < s implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r )
assume that
A16: x1 in dom (f | X) and
A17: abs (x1 - x0) < s ; :: thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
B0: x0 in REAL by XREAL_0:def 1;
B1: x1 in REAL by XREAL_0:def 1;
||.(((f | X) /. x1) - ((f | X) /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A12, PARTFUN2:15, B0
.= ||.((f /. x1) - (f /. x0)).|| by A16, PARTFUN2:15, B1 ;
hence ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A11, A15, A16, A17; :: thesis: verum

end;

hence f | X is_continuous_in x0 by Th8, A12; :: thesis: verum

end;

hence f | X is continuous by Def2; :: thesis: verum