What type of interval should it be

D

Darya

Junior Member
Joined
Jan 17, 2020
Messages
154
Hi! I've got a problem with the following problem:
If a continuous function f, defined on the interval I is not bounded above, then..
a) I isn't bounded above
b) I is an open interval
c) I isn't a closed interval
d) none of the previous answers

I answered that "I isn't a closed interval" because if I is a closed interval and f is continuous, then f is bounded. Where am I mistaken?
THanks.
 
Darya said:
Hi! I've got a problem with the following problem:
If a continuous function f, defined on the interval I is not bounded above, then..
a) I isn't bounded above
b) I is an open interval
c) I isn't a closed interval
d) none of the previous answers

I answered that "I isn't a closed interval" because if I is a closed interval and f is continuous, then f is bounded. Where am I mistaken?
THanks.
Click to expand...
Why do you think you are mistaken?
 
The theorem being referenced here is
"If a function is continuous on a closed and bounded interval then it is bounded".

If a continuous function is not bounded then the interval is either not bounded or not closed.

For example, the interval \(\displaystyle [0,\infty)\) is closed but not bounded and the continuous function f(x)= x is not bounded above on it.

Conversely, the iterval \(\displaystyle (0, 1)\) is bounded but not closed and the continuous function f(x)= 1/x is not bounded above on it.
 
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