let f be PartFunc of REAL,REAL; :: thesis: ( f is total & ( for x1, x2 being Real holds f . (x1 + x2) = (f . x1) + (f . x2) ) & ex x0 being Real st f is_continuous_in x0 implies f | REAL is continuous )
assume that
A1: f is total and
A2: for x1, x2 being Real holds f . (x1 + x2) = (f . x1) + (f . x2) ; :: thesis: ( for x0 being Real holds not f is_continuous_in x0 or f | REAL is continuous )
A3: dom f = REAL by A1, PARTFUN1:def 2;
given x0 being Real such that A4: f is_continuous_in x0 ; :: thesis: f | REAL is continuous
A5: (f . x0) + 0 = f . (x0 + 0)
.= (f . x0) + (f . 0) by A2 ;

A6: now :: thesis: for x1 being Real holds - (f . x1) = f . (- x1)

let x1 be Real; :: thesis: - (f . x1) = f . (- x1)
0 = f . (x1 + (- x1)) by A5
.= (f . x1) + (f . (- x1)) by A2 ;
hence - (f . x1) = f . (- x1) ; :: thesis: verum

end;

A7: now :: thesis: for x1, x2 being Real holds f . (x1 - x2) = (f . x1) - (f . x2)

let x1, x2 be Real; :: thesis: f . (x1 - x2) = (f . x1) - (f . x2)
thus f . (x1 - x2) = f . (x1 + (- x2))
.= (f . x1) + (f . (- x2)) by A2
.= (f . x1) + (- (f . x2)) by A6
.= (f . x1) - (f . x2) ; :: thesis: verum

end;

now :: thesis: for x1, r being Real st x1 in REAL & r > 0 holds
ex s being Real st
( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds
|.((f . x2) - (f . x1)).| < r ) )

let x1, r be Real; :: thesis: ( x1 in REAL & r > 0 implies ex s being Real st
( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds
|.((f . x2) - (f . x1)).| < r ) ) )
assume that
x1 in REAL and
A8: r > 0 ; :: thesis: ex s being Real st
( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds
|.((f . x2) - (f . x1)).| < r ) )
set y = x1 - x0;
consider s being Real such that
A9: 0 < s and
A10: for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
|.((f . x1) - (f . x0)).| < r by A4, A8, Th3;
take s = s; :: thesis: ( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds
|.((f . x2) - (f . x1)).| < r ) )
thus s > 0 by A9; :: thesis: for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds
|.((f . x2) - (f . x1)).| < r
let x2 be Real; :: thesis: ( x2 in REAL & |.(x2 - x1).| < s implies |.((f . x2) - (f . x1)).| < r )
assume that
x2 in REAL and
A11: |.(x2 - x1).| < s ; :: thesis: |.((f . x2) - (f . x1)).| < r
A12: ( x2 - (x1 - x0) in REAL & |.((x2 - (x1 - x0)) - x0).| = |.(x2 - x1).| ) by XREAL_0:def 1;
(x1 - x0) + x0 = x1 ;
then |.((f . x2) - (f . x1)).| = |.((f . x2) - ((f . (x1 - x0)) + (f . x0))).| by A2
.= |.(((f . x2) - (f . (x1 - x0))) - (f . x0)).|
.= |.((f . (x2 - (x1 - x0))) - (f . x0)).| by A7 ;
hence |.((f . x2) - (f . x1)).| < r by A3, A10, A11, A12; :: thesis: verum

end;

hence f | REAL is continuous by A3, Th14; :: thesis: verum