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 be PartFunc of S,S st f = id dom f holds f is_continuous_on dom f
proof
  let f be PartFunc of S,S;
  assume
A1: f = id dom f;
  now
    let x0 such that
A2: x0 in dom f;
    thus f/.x0=f.x0 by A2,PARTFUN1:def 6
      .= x0 by A1,A2,FUNCT_1:17;
  end;
  hence thesis by Th49;
end;
