reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem
  for f st (ex r be Point of T st rng f = {r}) holds f is_continuous_on (dom f)
proof
  let f;
  given r be Point of T such that
A1: rng f = {r};
  now
    let x1,x2;
    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.T .|| by A4,RLVECT_1:15
      .= 0 by NORMSP_1:1;
    hence ||. f/.x1-f/.x2.|| <= 1*||. x1-x2.|| by NORMSP_1:4;
  end;
  then f is_Lipschitzian_on dom f;
  hence thesis by Th45;
end;
