Let . For each interval, determine if we can use the Boundedness Theorem to conclude that f must be bounded on that interval.
If not, explain why not.
[0, 1]
[1, 2]
(0, 1)
(0, 1]
[0, 1]: We can't use the Boundedness Theorem, because one of the assumptions fails: f is not continuous on the interval [0, 1], since f is undefined at x = 0.
[1, 2]: We can use the Boundedness Theorem to conclude that f is bounded on [1, 2] because f is continuous on [1, 2] and this interval is closed.
(0, 1): We can't use the Boundedness Theorem, because this interval is not closed.
(0, 1] : We can't use the Boundedness Theorem, because this interval is not closed.
. For each interval, determine if we can use the Boundedness Theorem to conclude that f must be bounded on that interval.