Let be the square . The function is continous everywhere because entire functions (differenciable) are continous. Thus, is bounded on because it is a closed and bounded set. But then is bounded by that same number because the value of is determined by within the rectangle. So for example, . Thus, any value outside the rectangle is known from the value inside the rectangle. Since the function is bounded on this rectangle is is bounded outside it too because it has the same values as inside the rectangle.