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;

theorem
  for f be PartFunc of RNS,CNS, x0 be Point of RNS st x0 in dom f & (ex
  N be Neighbourhood of x0 st dom f /\ N = {x0}) holds f is_continuous_in x0
proof
  let f be PartFunc of RNS,CNS;
  let x0 be Point of RNS;
  assume
A1: x0 in dom f;
  given N be Neighbourhood of x0 such that
A2: dom f /\ N = {x0};
  now
    let N1 be Neighbourhood of f/.x0;
    take N;
A3: f/.x0 in N1 by Th3;
    f.:N = Im(f,x0) by A2,RELAT_1:112
      .= {f.x0} by A1,FUNCT_1:59
      .= {f/.x0} by A1,PARTFUN1:def 6;
    hence f.:N c= N1 by A3,ZFMISC_1:31;
  end;
  hence thesis by A1,Th19;
end;
