Does a function that is bounded on [a, b] need to be continuous on [a, b]? Why or why not?
Answer
No. A function can be bounded on [a, b] but discontinuous on [a, b]. Here's an example:
Example 2
Does a function that is continuous on an open interval (a, b) need to be bounded on that interval?
Answer
No. Here's an example:
With open intervals, there's plenty of room for continuous functions to approach infinity as they approach the endpoints of the interval.
Example 3
Does a function that is discontinuous on a closed interval [a, b] need to be bounded on that interval?
Answer
Nope. Here's an example.
Not only does this function have a vertical asymptote at x = a, it isn't even defined at x = a to begin with.
Example 4
Let . For each given function and interval, determine if we can use the Boundedness Theorem to conclude the function is bounded on that interval. If not, explain why not.
[0, 5]
(0, 5)
(2, 3)
[0, 1]
[0, 2)
Answer
[0, 5]: We cannot use the Boundedness Theorem, because f is not continuous on this interval (f is not continuous at x = 2).
(0, 5): We cannot use the Boundedness Theorem. Both assumptions fail, since f is not continuous on this interval, and the interval is not closed.
(2, 3): We cannot use the Boundedness Theorem. This interval is not closed.
[0, 1]: For a change, we can use the Boundedness Theorem. This interval is closed and the function f is continuous on [0, 1].
[0, 2): We cannot use the Boundedness Theorem. This interval is not closed.
. For each given function and interval, determine if we can use the Boundedness Theorem to conclude the function is bounded on that interval. If not, explain why not.