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 for r be Real for p be Point of S holds (X c=
dom f & (for x0 st x0 in X holds f/.x0 = r*x0+p) implies f is_continuous_on X )
proof
  let f be PartFunc of S,S;
  let r be Real;
  let p be Point of S;
  assume that
A1: X c= dom f and
A2: for x0 st x0 in X holds f/.x0 = r*x0+p;
  now
    0+0<|.r.|+1 by COMPLEX1:46,XREAL_1:8;
    hence 0<|.r.|+1;
    let x1,x2 be Point of S;
    assume x1 in X & x2 in X;
    then f/.x1 = r*x1+p & f/.x2 = r*x2+p by A2;
    then
A3: ||. f/.x1-f/.x2.|| = ||. r*x1+(p-(p+r*x2)).|| by RLVECT_1:def 3
      .= ||. r*x1+(p-p-r*x2).|| by RLVECT_1:27
      .= ||. r*x1+(0.S-r*x2).|| by RLVECT_1:15
      .= ||. r*x1-r*x2.||
      .= ||. r*(x1-x2).|| by RLVECT_1:34
      .= |.r.|*||. x1-x2.|| by NORMSP_1:def 1;
    0<=||. x1-x2.|| by NORMSP_1:4;
    then
    ||. f/.x1-f/.x2.|| + 0 <= |.r.|*||. x1-x2.|| + 1*||. x1-x2.|| by A3,
XREAL_1:7;
    hence ||. f/.x1-f/.x2.|| <= (|.r.|+1)*||. x1-x2.||;
  end;
  then f is_Lipschitzian_on X by A1;
  hence thesis by Th45;
end;
