Boundedness Examples

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.