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# Quiz: Continuity

last edited by 11 years, 8 months ago

## This quiz is designed to test your knowledge of limits of functions and of continuity and uniform continuity.

Discuss this quiz

(Key: correct, incorrect, partially correct.)

1. Let  be a subset of . What does it mean for  to be an adherent point of ?
1. For every , there exists a  in  such that .
• CORRECT.
2. For every , there exists a  in  such that .
• INCORRECT. This is what it means for x to be a limit point of , not an adherent point.
3. There exists  and  such that .
4. For every  and all , we have .
5. For every , there exists  such that .
6. There exists  such that  for all  in .
7. There exists  such that  for all .
2. Let  be a subset of . What does it mean for  to not be an adherent point of ?
1. There exists an  such that  for all  in .
• CORRECT.
2. There exists an  and  such that .
3. For every  there exists   such that .
4. For every  we have  for every .
5. For every  there exists an  such that .
6. There exists   such that  for every .
7. There exists an  such that  whenever .
3. Let  be a subset of . What does it mean for  to be a limit point of ?
1. For every , there exists a  such that .
• INCORRECT. This is what it means for  to be an adherent point of , not a limit point.
2. For every , there exists a  such that .
• CORRECT.
3. There exists  and   such that .
4. For every  and all , we have .
5. For every , there exists  such that .
6. There exists  such that  for all .
7. There exists  such that  for all .
4. Let  be a subset of . What does it mean for  to be an adherent point of ?
1. There exists a sequence  in  which converges to .
• CORRECT.
2. There exists a sequence  in  which converges to .
• INCORRECT. This is what it means for  to be a limit point of , not an adherent point.
3. Every sequence  in  which converges, converges to .
4. Every sequence  in  converges to .
5. Every Cauchy sequence  in  converges to .
6. There exists a sequence  in  which converges to .
7. Every sequence  which converges to , must lie in .
5. Let  be a subset of . What does it mean for  to be a limit point of ?
1. There exists a sequence  in  which converges to .
• INCORRECT. This is what it means for  to be an adherent point of , not a limit point.
2. There exists a sequence  of points in which converges to .
• CORRECT.
3. Every sequence  in  which converges, converges to .
4. Every sequence  in  converges to .
5. Every Cauchy sequence  in  converges to .
6. Every sequence  which converges to , must lie in .
7. Every sequence  which converges to , must lie in .
6. Let  be a subset of , let  be a function, and let  be an adherent point of . What does it mean for  to equal ?
1. For every , there exists a  such that for all  for which .
2. For every , there exists a  such that  for all  for which .
3. For every , there exists a  such that  for all  for which .
• CORRECT.
4. For every  and , we have  for all  for which .
5. For every  and every , there exists  such that  if .
6. For every , there exists a  such that  and  for some .
7. For every , there exists a  such that  and  for all .
7. Let  be a subset of , let  be a function, and let  be an element of . What does it mean for  to be continuous at ?
1. For every , there exists a  such that  for all  for which .
2. For every , there exists a  such that  for all  for which .
3. For every , there exists a  such that  for all  for which .
• CORRECT.
4. For every  and , we have  for all  for which .
5. For every  and every , there exists  such that  if .
6. For every , there exists a  such that  and  for some .
7. For every , there exists a  such that  and  for all .
8. Let  be a subset of , let  be a function, and let  be an adherent point of . What does it mean for to equal ?
1. For every sequence  in  which converges to , the sequence  converges to .
• CORRECT.
2. There exists a sequence  in  converging to , such that  converges to .
3. Whenever  converges to , the sequence , must then converge to .
4. For every sequence  in  which converges to , the sequence  converges to .
5. Whenever  converges to , the sequence , must lie in  and converge to .
6. Whenever  converges to , the sequence , must lie in  and converge to .
7. For every sequence  in  which converges to L, the sequence  converges to .
9. Let  be a subset of , let  be a function, and let  be an element of . What does it mean for  to be continuous at ?
1. For every sequence  in  which converges to , the sequence  converges to .
• CORRECT.
2. There exists a sequence  in  converging to , such that  converges to .
3. Whenever  converges to , the sequence , must then converge to .
4. Every sequence  which is convergent, converges to .
5. Whenever  converges to f(x), the sequence , must lie in  and converge to .
6. Whenever  converges to , the sequence  must lie in  and converge to .
7. For every sequence  in , the sequence  converges to .
10. Let  be a subset of , let  be a function. What does it mean for  to be continuous on ?
1. For every epsilon > 0 and  in X, there exists a delta > 0 such that  for all  for which .
• CORRECT.
2. For every , there exists a  such that  for all x, y in  for which .
• INCORRECT. This is what it means for  to be uniformly continuous on .
3. For every , there exists a  and x,y in  such that which  and .
4. For every  and , there exists a  such that  for all y in X for which |y-x| < epsilon.
5. For every , there exists an  and  such that  and .
6. For every  and , there exists a  such that  for all  for which .
7. For every , there exists a  such that  for all  for which .
11. Let  be a subset of , let  be a function. What does it mean for  to be uniformly continuous on ?
1. For every  and , there exists a  such that which  for all  for which .
• INCORRECT. This is what it means for  to be continuous on .
2. For every , there exists a  such that  for all  for which .
• CORRECT.
3. For every , there exists a  and x,y in  such that  and .
4. For every  and  in , there exists a  such that  for all  for which .
5. For every , there exists an  and  such that  and .
6. For every  and  in , there exists a  such that  for all  for which .
7. For every , there exists a  such that  for all  for which .
12. Let  be a continuous function such that  and . The most we can say about the set  is that
1. It is a closed interval which contains [3,6].
• CORRECT.
2. It is a set which contains [3,6].
3. It is the interval [3,6].
4. It is a closed interval.
5. It is the interval [2,4].
6. It is a set which contains 3 and 6.
7. It is a bounded set.
13. Let  be a continuous function such that  and . The most we can say about the set  is that
1. It is a set which contains [3,6].
• PARTIALLY.  This is true, but more can be said about f.
2. It is a bounded set which contains [3,6].
• INCORRECT.  f need not be bounded.
3. It is the interval [3,6].
4. It is an open interval which contains [3,6].
• INCORRECT.  The set need not be open.
5. It is an interval which contains [3,6].
• CORRECT.
6. It is a bounded interval which contains [3,6].
• INCORRECT.  f need not be bounded.
7. It is a set with contains both 3 and 6.
• INCORRECT.  Much more can be said about f!
14. Let  be a uniformly continuous function such that  and . The most we can say about the set  is that
1. It is a set which contains [3,6].
2. It is a bounded set which contains [3,6].
• PARTIALLY.  This is correct, but more can be said about f.
3. It is an interval which contains [3,6].
• PARTIALLY.  This is correct, but more can be said about f.
4. It is an open bounded interval which contains [3,6].
• INCORRECT.  The set need not be open.
5. It is a bounded interval which contains [3,6].
• CORRECT.
6. It is an open interval which contains [3,6].
• INCORRECT.  The set need not be open.
7. It is a bounded set.

## Score:

#### Kevin Ventullo said

at 6:30 pm on Dec 25, 2008

On a couple of the questions, the TeX doesn't show up.

Also, shouldn't the respective answers to 13 and 14 be "an interval which contains [3,6]" and "a bounded interval which contains [3,6]"?

#### Terence Tao said

at 7:08 pm on Dec 25, 2008

Hmm, the TeX seems to work fine in my browser...

Thanks for pointing out the corrections in Q13 and Q14! I've adjusted the answers (and added some more commentary) accordingly.

#### test said

at 9:46 pm on Dec 25, 2008

Regarding TeX not showing: it might just be the server not serving the images as it should. Were there any broken images, or errors while loading the page? Did a reload correct it?

#### RH said

at 9:50 pm on Dec 25, 2008

(Above comment was by me, and "Delete" is broken. Sorry.)

#### Kevin Ventullo said

at 10:39 pm on Dec 25, 2008

Actually, there is just one missing image:

Question 4, one of the incorrect choices reads:
----
There exists a sequence x_1, x_2, x_3,... in which converges to x.
----
I'm guessing it should display X/{x} between "in" and "which".

Question 5 doesn't have a TeX problem, but there seems to be a typo in the correct choice:
----
There exists a sequence x_1, x_2, x_3,... in X which converges to x.
----
As opposed to
There exists a sequence x_1, x_2, x_3,... in X\{x} which converges to x.

I fixed a few other typos as well (missing or extra characters), I hope you don't mind.

#### RH said

at 4:03 pm on Dec 27, 2008

Corrections are always welcome. Thanks!

#### Alexei said

at 11:42 am on Jan 6, 2009

There are two identical answers for Q5.

#### Terence Tao said

at 8:01 pm on Jan 6, 2009

Oops! It should be fixed now, thanks!

#### almagest said

at 7:23 pm on Sep 28, 2019

How do I edit this page? [the intro line to Q6 needs f(y) instead of f(x)). I changed my password (haven't been here for years), but math bits just appear as coloured blobs. So presumably not mathjax. Don't see how to edit it.