theorem
  f is_continuous_on X & (f|X)"{0} = {} implies f^ is_continuous_on X
proof
  assume that
A1: f is_continuous_on X and
A2: (f|X)"{0} = {};
  f|X is_continuous_on X by A1,Th40;
  then (f|X)^ is_continuous_on X by A2,Th47;
  then (f^)|X is_continuous_on X by CFUNCT_1:54;
  hence thesis by Th40;
end;
