reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  for f st (ex r st rng f = {r}) holds f is continuous
proof
  let f;
  given r 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
A4: f.x2=r by A1,TARSKI:def 1;
    f.x1 in rng f by A2,FUNCT_1:def 3;
    then f.x1=r by A1,TARSKI:def 1;
    then |.f.x1-f.x2.| = 0 by A4,ABSVALUE:2;
    hence |.f.x1-f.x2.| <= 1*|.x1-x2.| by COMPLEX1:46;
  end;
  then f is Lipschitzian;
  hence thesis;
end;
