reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th75:
  for T being non empty TopSpace for f being Function of T, I[01]
holds f is continuous iff for a,b being Real st 0 <= a & a < 1 & 0 < b &
  b <= 1 holds f"[.0,b.[ is open & f"].a,1.] is open
proof
  set A3 = {].a,b.[ where a,b is Real: 0 <= a & a < b & b <= 1};
  set A2 = {].a,1.] where a is Real: 0 <= a & a < 1};
  set A1 = {[.0,a.[ where a is Real: 0 < a & a <= 1};
  reconsider B = A1 \/ A2 \/ A3 as Basis of I[01] by Th74;
  let T be non empty TopSpace;
  let f be Function of T, I[01];
  hereby
    assume
A1: f is continuous;
    let a,b be Real;
    reconsider x = a, y = b as Real;
    assume that
A2: 0 <= a and
A3: a < 1 and
A4: 0 < b and
A5: b <= 1;
    ].x,1.] in A2 by A2,A3;
    then ].x,1.] in A1 \/ A2 by XBOOLE_0:def 3;
    then
A6: ].x,1.] in B by XBOOLE_0:def 3;
    [.0,y.[ in A1 by A4,A5;
    then [.0,y.[ in A1 \/ A2 by XBOOLE_0:def 3;
    then [.0,y.[ in B by XBOOLE_0:def 3;
    hence f"[.0,b.[ is open & f"].a,1.] is open by A6,A1,YELLOW_9:34;
  end;
  assume
A7: for a,b being Real st 0 <= a & a < 1 & 0 < b & b <= 1 holds
  f"[.0,b.[ is open & f"].a,1.] is open;
  now
    let A be Subset of I[01];
    assume A in B;
    then
A8: A in A1 \/ A2 or A in A3 by XBOOLE_0:def 3;
    per cases by A8,XBOOLE_0:def 3;
    suppose
      A in A1;
      then ex a being Real st A = [.0,a.[ & 0 < a & a <= 1;
      hence f"A is open by A7;
    end;
    suppose
      A in A2;
      then ex a being Real st A = ].a,1.] & 0 <= a & a < 1;
      hence f"A is open by A7;
    end;
    suppose
      A in A3;
      then consider a,b being Real such that
A9:   A = ].a,b.[ and
A10:  0 <= a and
A11:  a < b and
A12:  b <= 1;
      a < 1 by A11,A12,XXREAL_0:2;
      then reconsider
      U = f"[.0,b.[, V = f"].a,1.] as open Subset of T by A10,A11,A7,A12;
      A = [.0,b.[ /\ ].a,1.] by A9,A10,A12,XXREAL_1:153;
      then f"A = U /\ V by FUNCT_1:68;
      hence f"A is open;
    end;
  end;
  hence thesis by YELLOW_9:34;
end;
