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.
Make it rain.
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