That is correct. The Extreme Value Theorem says that if f (a real-valued function of a real variable) is continuous on a closed bounded interval, then it achieves its maximum and minimum value. Since your interval is not closed, the theorem says nothing about f, so any behavior is consistent with the theorem.
Your function achieves both maximum and minimum. If the interval were (-1,1), it would achieve its minimum but not its maximum. And if the interval were (0,1) it would achieve neither.