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, 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 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, 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, the carrier of S; :: 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: x0 in REAL by XREAL_0:def 1;
A10: x1 in REAL by XREAL_0:def 1;
A11: 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, A11, A7, A8; :: thesis: verum
end;
assume A12: 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 A13: x0 in dom (f | X) ; :: thesis:
then A14: x0 in X ;
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
A15: 0 < s and
A16: 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 A15; :: 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
A17: x1 in dom (f | X) and
A18: |.(x1 - x0).| < s ; :: thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
A19: x0 in REAL by XREAL_0:def 1;
A20: x1 in REAL by XREAL_0:def 1;
||.(((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 ; :: thesis: verum
end;
hence f | X is continuous ; :: thesis: verum