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

proof

assume A2: f | X is continuous ; :: thesis: for x0, r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Real st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
let x0, r be Real; :: thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Real st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1 being Real st x1 in X & |.(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;
then consider s being Real such that
A5: 0 < s and
A6: for x1 being Real st x1 in dom (f | X) & |.(x1 - x0).| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r by A4, Th3;
take s ; :: thesis: ( 0 < s & ( for x1 being Real st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus 0 < s by A5; :: thesis: for x1 being Real st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r
let x1 be Real; :: thesis: ( x1 in X & |.(x1 - x0).| < s implies |.((f /. x1) - (f /. x0)).| < r )
assume that
A7: x1 in X and
A8: |.(x1 - x0).| < s ; :: thesis: |.((f /. x1) - (f /. x0)).| < r
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
.= |.(((f | X) /. x1) - ((f | X) /. x0)).| by A3, A9, PARTFUN2:15 ;
hence |.((f /. x1) - (f /. x0)).| < r by A6, A9, A7, A8; :: thesis: verum

end;

assume A10: for x0, r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Real st x1 in X & |.(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 :: thesis: for x0 being Real st x0 in dom (f | X) holds
f | X is_continuous_in x0

let x0 be Real; :: 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
A13: x0 in X by A12, RELAT_1:57;
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Real st x1 in dom (f | X) & |.(x1 - x0).| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r ) )

proof

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

end;

hence f | X is_continuous_in x0 by Th3, A12; :: thesis: verum

end;

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