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;

theorem Th60:
  for A being Subset of X, x being Point of X, x0 being Point of
  X0 st A is open & x in A & A c= the carrier of X0 & x = x0 holds f
  is_continuous_at x iff f|X0 is_continuous_at x0
proof
  let A be Subset of X, x be Point of X, x0 be Point of X0 such that
A1: A is open & x in A and
A2: A c= the carrier of X0 and
A3: x = x0;
  thus f is_continuous_at x implies f|X0 is_continuous_at x0 by A3,Th58;
  thus f|X0 is_continuous_at x0 implies f is_continuous_at x
  proof
    assume
A4: f|X0 is_continuous_at x0;
    A is a_neighborhood of x by A1,CONNSP_2:3;
    hence thesis by A2,A3,A4,Th59;
  end;
end;
