theorem
  f is_differentiable_on X implies f|X is continuous
proof
  assume A1: f is_differentiable_on X;
  reconsider g=f as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
A2: X c= dom g by A1;
  now
    let x;
    assume x in X;
    then f|X is_differentiable_in x by A1;
    hence g|X is_differentiable_in x;
  end;
  then g is_differentiable_on X by A2,NDIFF_3:def 5;
  then g|X is continuous by NDIFF_3:23;
  hence thesis by NFCONT_4:23;
end;
