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
  for f be PartFunc of CNS1,CNS2 st (ex r be Point of CNS2 st rng f = {r
  }) holds f is_continuous_on dom f
proof
  let f be PartFunc of CNS1,CNS2;
  given r be Point of CNS2 such that
A1: rng f = {r};
  now
    let x1,x2 be Point of CNS1;
    assume that
A2: x1 in dom f and
A3: x2 in dom f;
    f.x2 in rng f by A3,FUNCT_1:def 3;
    then f/.x2 in rng f by A3,PARTFUN1:def 6;
    then
A4: f/.x2=r by A1,TARSKI:def 1;
    f.x1 in rng f by A2,FUNCT_1:def 3;
    then f/.x1 in rng f by A2,PARTFUN1:def 6;
    then f/.x1=r by A1,TARSKI:def 1;
    then ||. f/.x1-f/.x2.|| = ||. 0.CNS2 .|| by A4,RLVECT_1:15
      .= 0 by CLVECT_1:102;
    hence ||. f/.x1-f/.x2.|| <= 1*||. x1-x2.|| by CLVECT_1:105;
  end;
  then f is_Lipschitzian_on dom f;
  hence thesis by Th116;
end;
