Continuity of Functions Exercises

Answer

No. A function can be bounded on [a, b] but discontinuous on [a, b]. Here's an example:

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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.

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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.

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.