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

now :: thesis: for x0 being real number st x0 in dom (f | X) holds
(f | X) . x0 = abs x0

let x0 be real number ; :: thesis: ( x0 in dom (f | X) implies (f | X) . x0 = abs x0 )
assume A2: x0 in dom (f | X) ; :: thesis: (f | X) . x0 = abs x0
then f . x0 = abs x0 by A1;
hence (f | X) . x0 = abs x0 by A2, FUNCT_1:47; :: thesis: verum

end;

hence f | X is continuous by Th44; :: thesis: verum