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;

theorem Th44:
  f is continuous iff for x being Point of X holds f is_continuous_at x
proof
  thus f is continuous implies for x being Point of X holds f is_continuous_at
  x;
  assume
A1: for x being Point of X holds f is_continuous_at x;
  thus f is continuous
  proof
    let x be Point of X, G be a_neighborhood of f.x;
    f is_continuous_at x by A1;
    hence thesis;
  end;
end;
