let f be PartFunc of REAL,REAL; :: thesis: ( ( for x0 being real number st x0 in dom f holds
f . x0 = x0 ) implies f is continuous )
assume A1: for x0 being real number st x0 in dom f holds
f . x0 = x0 ; :: thesis: f is continuous

now

let x1, x2 be real number ; :: thesis: ( x1 in dom f & x2 in dom f implies abs ((f . x1) - (f . x2)) <= 1 * (abs (x1 - x2)) )
assume that
A2: x1 in dom f and
A3: x2 in dom f ; :: thesis: abs ((f . x1) - (f . x2)) <= 1 * (abs (x1 - x2))
f . x1 = x1 by A1, A2;
hence abs ((f . x1) - (f . x2)) <= 1 * (abs (x1 - x2)) by A1, A3; :: thesis: verum

end;

then f is Lipschitzian by Def3;
hence f is continuous ; :: thesis: verum