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 Th49:
  for f be PartFunc of S,S st (for x0 st x0 in dom f holds f/.x0 =
  x0) holds f is_continuous_on dom f
proof
  let f be PartFunc of S,S such that
A1: for x0 st x0 in dom f holds f/.x0 = x0;
  now
    let x1,x2;
    assume that
A2: x1 in dom f and
A3: x2 in dom f;
    f/.x1 = x1 by A1,A2;
    hence ||. f/.x1-f/.x2.|| <= 1*||. x1-x2.|| by A1,A3;
  end;
  then f is_Lipschitzian_on dom f;
  hence thesis by Th45;
end;
