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

theorem
(ex r be Point of S st rng f = {r}) implies f is continuous
proof
   given r be Point of S such that A1: rng f = {r};
   now
    let x1,x2;
    assume A2: x1 in dom f & x2 in dom f; then
    f.x2 in rng f by FUNCT_1:def 3; then
    f/.x2 in rng f by A2,PARTFUN1:def 6; then
A3: 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 by A3,NORMSP_1:6;
    hence ||. f/.x1-f/.x2 .|| <= 1*|.x1-x2.| by COMPLEX1:46;
   end; then
   f is Lipschitzian;
   hence thesis;
end;
