let X be set ; :: thesis: for f being PartFunc of REAL,REAL 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; :: 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 ;
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 ;
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 ;
then |.((f . x1) - (f . x0)).| = |.(((f | X) . x1) - (f . x0)).| by
.= |.(((f | X) . x1) - ((f | X) . x0)).| by ;
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
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 A11: x0 in dom (f | X) ; :: thesis:
A12: x0 in X by A11;
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
A13: 0 < s and
A14: for x1 being Real st x1 in X & |.(x1 - x0).| < s holds
|.((f . x1) - (f . x0)).| < r by ;
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 A13; :: 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
A15: x1 in dom (f | X) and
A16: |.(x1 - x0).| < s ; :: thesis: |.(((f | X) . x1) - ((f | X) . x0)).| < r
|.(((f | X) . x1) - ((f | X) . x0)).| = |.(((f | X) . x1) - (f . x0)).| by
.= |.((f . x1) - (f . x0)).| by ;
hence |.(((f | X) . x1) - ((f | X) . x0)).| < r by ; :: thesis: verum
end;
hence f | X is_continuous_in x0 by Th3; :: thesis: verum
end;
hence f | X is continuous ; :: thesis: verum