
theorem Th80b:
  for X be strict non empty SubSpace of R^1,
      f be RealMap of X,
      g be PartFunc of REAL,REAL
    st g = f
  holds
    f is continuous iff g is continuous
proof
  let X be strict non empty SubSpace of R^1,
      f be RealMap of X,
      g be PartFunc of REAL,REAL;
  assume A1: g=f;
A2: dom g = the carrier of X by FUNCT_2:def 1,A1;
  hereby
    assume B3: f is continuous;
    for x0 being Real st x0 in dom g holds g is_continuous_in x0
    proof
      let x0 be Real;
      assume x0 in dom g; then
      reconsider x=x0 as Point of X by FUNCT_2:def 1,A1;
      for V be Subset of REAL st f.x in V & V is open holds
        ex W be Subset of X st
          x in W & W is open & f.: W c= V by B3,C0SP2:1;
      hence g is_continuous_in x0 by A1,Th80a;
    end;
    hence g is continuous by FCONT_1:def 2;
  end;
  assume B5: g is continuous;
  for x be Point of X
    for V be Subset of REAL st f.x in V & V is open holds
      ex W be Subset of X st
        x in W & W is open & f.:W c= V
  proof
    let x be Point of X;
    let V be Subset of REAL;
    assume A6: f.x in V & V is open;
    reconsider x0=x as Real;
    g is_continuous_in x0 by B5,FCONT_1:def 2,A2;
    hence ex W be Subset of X st
            x in W & W is open & f.: W c= V by A1,A6,Th80a;
  end;
  hence f is continuous by C0SP2:1;
end;
