reserve n,m for Element of NAT;
reserve r,s for Real;
reserve z for Complex;
reserve CNS,CNS1,CNS2 for ComplexNormSpace;
reserve RNS for RealNormSpace;
reserve X,X1 for set;

theorem Th128:
  for f be PartFunc of CNS,CNS st (for x0 be Point of CNS st x0
  in dom f holds f/.x0 = x0) holds f is_continuous_on dom f
proof
  let f be PartFunc of CNS,CNS such that
A1: for x0 be Point of CNS st x0 in dom f holds f/.x0 = x0;
  now
    let x1,x2 be Point of CNS;
    assume that
A2: x1 in dom f and
A3: x2 in dom f;
    f/.x1 = x1 by A1,A2;
    hence ||. f/.x1-f/.x2.|| <= 1*||. x1-x2.|| by A1,A3;
  end;
  then f is_Lipschitzian_on dom f;
  hence thesis by Th116;
end;
