let X be set ; :: thesis: for f being PartFunc of REAL,REAL st X c= dom f & ( for x0 being real number st x0 in X holds
f . x0 = x0 ^2 ) holds
f | X is continuous
let f be PartFunc of REAL,REAL; :: thesis: ( X c= dom f & ( for x0 being real number st x0 in X holds
f . x0 = x0 ^2 ) implies f | X is continuous )
assume that
A1: X c= dom f and
A2: for x0 being real number st x0 in X holds
f . x0 = x0 ^2 ; :: thesis: f | X is continuous
X = (dom f) /\ X by A1, XBOOLE_1:28;
then A3: X = dom (f | X) by RELAT_1:90;

now

let x0 be real number ; :: thesis: ( x0 in dom (f | X) implies (f | X) . x0 = x0 ^2 )
assume A4: x0 in dom (f | X) ; :: thesis: (f | X) . x0 = x0 ^2
then f . x0 = x0 ^2 by A2, A3;
hence (f | X) . x0 = x0 ^2 by A4, FUNCT_1:70; :: thesis: verum

end;

then (f | X) | X is continuous by A3, Th49;
hence f | X is continuous by RELAT_1:101; :: thesis: verum