theorem
  f is_differentiable_on X implies f is_continuous_on X
proof
  assume
A1: f is_differentiable_on X;
  hence
A2: X c= dom f;
  let x be Complex;
  assume x in X;
  then
A3: x in dom (f|X) by A2,RELAT_1:57;
  f|X is differentiable by A1;
  then f|X is_differentiable_in x by A3;
  hence thesis by Th34;
end;
