No. A function can be bounded on [a, b] but discontinuous on [a, b]. Here's an example:
Does a function that is continuous on an open interval (a, b) need to be bounded on that interval?
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.
Does a function that is discontinuous on a closed interval [a, b] need to be bounded on that interval?
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.
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.
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