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 Th44:
  (for x0 st x0 in dom f holds f.x0 = |.x0.|) implies f is continuous
proof
  assume
A1: for x0 st x0 in dom f holds f.x0 = |.x0.|;
  now
    let x1,x2;
    assume x1 in dom f & x2 in dom f;
    then f.x1 = |.x1.| & f.x2 = |.x2.| by A1;
    hence |.f.x1-f.x2.| <= 1*|.x1-x2.| by COMPLEX1:64;
  end;
  then f is Lipschitzian;
  hence thesis;
end;
