reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;
reserve X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;

theorem
  for f being Function of X,Y, X1, X2 being non empty SubSpace of X st X
  = X1 union X2 for x being Point of X, x1 being Point of X1, x2 being Point of
X2 st x = x1 & x = x2 holds f is_continuous_at x iff f|X1 is_continuous_at x1 &
  f|X2 is_continuous_at x2
proof
  let f be Function of X,Y, X1, X2 be non empty SubSpace of X such that
A1: X = X1 union X2;
  let x be Point of X, x1 be Point of X1, x2 be Point of X2;
  assume that
A2: x = x1 and
A3: x = x2;
  thus f is_continuous_at x implies f|X1 is_continuous_at x1 & f|X2
  is_continuous_at x2 by A2,A3,Th58;
  thus f|X1 is_continuous_at x1 & f|X2 is_continuous_at x2 implies f
  is_continuous_at x
  proof
    assume that
A4: f|X1 is_continuous_at x1 and
A5: f|X2 is_continuous_at x2;
    for G being a_neighborhood of f.x ex H being a_neighborhood of x st f
    .:H c= G
    proof
      let G be a_neighborhood of f.x;
      f.x = (f|X1).x1 by A2,FUNCT_1:49;
      then consider H1 being a_neighborhood of x1 such that
A6:   (f|X1).:H1 c= G by A4;
      the carrier of X1 c= the carrier of X by BORSUK_1:1;
      then reconsider S1 = H1 as Subset of X by XBOOLE_1:1;
      f.x = (f|X2).x2 by A3,FUNCT_1:49;
      then consider H2 being a_neighborhood of x2 such that
A7:   (f|X2).:H2 c= G by A5;
      the carrier of X2 c= the carrier of X by BORSUK_1:1;
      then reconsider S2 = H2 as Subset of X by XBOOLE_1:1;
      f.:S2 c= G by A7,FUNCT_2:97;
      then
A8:   S2 c= f"G by FUNCT_2:95;
      consider H being a_neighborhood of x such that
A9:   H c= H1 \/ H2 by A1,A2,A3,Th16;
      take H;
      f.:S1 c= G by A6,FUNCT_2:97;
      then S1 c= f"G by FUNCT_2:95;
      then S1 \/ S2 c= f"G by A8,XBOOLE_1:8;
      then H c= f"G by A9,XBOOLE_1:1;
      hence thesis by FUNCT_2:95;
    end;
    hence thesis;
  end;
end;
