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Quiz: Continuity
Page history
last edited
by Terence Tao 15 years, 2 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.)
- Let be a subset of . What does it mean for to be an adherent point of ?
- For every , there exists a in such that .
- 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.
- There exists and such that .
- For every and all , we have .
- For every , there exists such that .
- There exists such that for all in .
- There exists such that for all .
- Let be a subset of . What does it mean for to not be an adherent point of ?
- There exists an such that for all in .
- There exists an and such that .
- For every there exists such that .
- For every we have for every .
- For every there exists an such that .
- There exists such that for every .
- There exists an such that whenever .
- Let be a subset of . What does it mean for to be a limit point of ?
- For every , there exists a such that .
- INCORRECT. This is what it means for to be an adherent point of , not a limit point.
- For every , there exists a such that .
- There exists and such that .
- For every and all , we have .
- For every , there exists such that .
- There exists such that for all .
- There exists such that for all .
- Let be a subset of . What does it mean for to be an adherent point of ?
- There exists a sequence in which converges to .
- 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.
- Every sequence in which converges, converges to .
- Every sequence in converges to .
- Every Cauchy sequence in converges to .
- There exists a sequence in which converges to .
- Every sequence which converges to , must lie in .
- Let be a subset of . What does it mean for to be a limit point of ?
- 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.
- There exists a sequence of points in which converges to .
- Every sequence in which converges, converges to .
- Every sequence in converges to .
- Every Cauchy sequence in converges to .
- Every sequence which converges to , must lie in .
- Every sequence which converges to , must lie in .
- Let be a subset of , let be a function, and let be an adherent point of . What does it mean for to equal ?
- For every , there exists a such that for all for which .
- For every , there exists a such that for all for which .
- For every , there exists a such that for all for which .
- For every and , we have for all for which .
- For every and every , there exists such that if .
- For every , there exists a such that and for some .
- For every , there exists a such that and for all .
- Let be a subset of , let be a function, and let be an element of . What does it mean for to be continuous at ?
- For every , there exists a such that for all for which .
- For every , there exists a such that for all for which .
- For every , there exists a such that for all for which .
- For every and , we have for all for which .
- For every and every , there exists such that if .
- For every , there exists a such that and for some .
- For every , there exists a such that and for all .
- Let be a subset of , let be a function, and let be an adherent point of . What does it mean for to equal ?
- For every sequence in which converges to , the sequence converges to .
- There exists a sequence in converging to , such that converges to .
- Whenever converges to , the sequence , must then converge to .
- For every sequence in which converges to , the sequence converges to .
- Whenever converges to , the sequence , must lie in and converge to .
- Whenever converges to , the sequence , must lie in and converge to .
- For every sequence in which converges to L, the sequence converges to .
- Let be a subset of , let be a function, and let be an element of . What does it mean for to be continuous at ?
- For every sequence in which converges to , the sequence converges to .
- There exists a sequence in converging to , such that converges to .
- Whenever converges to , the sequence , must then converge to .
- Every sequence which is convergent, converges to .
- Whenever converges to f(x), the sequence , must lie in and converge to .
- Whenever converges to , the sequence must lie in and converge to .
- For every sequence in , the sequence converges to .
- Let be a subset of , let be a function. What does it mean for to be continuous on ?
- For every epsilon > 0 and in X, there exists a delta > 0 such that for all for which .
- 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 .
- For every , there exists a and x,y in such that which and .
- For every and , there exists a such that for all y in X for which |y-x| < epsilon.
- For every , there exists an and such that and .
- For every and , there exists a such that for all for which .
- For every , there exists a such that for all for which .
- Let be a subset of , let be a function. What does it mean for to be uniformly continuous on ?
- For every and , there exists a such that which for all for which .
- INCORRECT. This is what it means for to be continuous on .
- For every , there exists a such that for all for which .
- For every , there exists a and x,y in such that and .
- For every and in , there exists a such that for all for which .
- For every , there exists an and such that and .
- For every and in , there exists a such that for all for which .
- For every , there exists a such that for all for which .
- Let be a continuous function such that and . The most we can say about the set is that
- It is a closed interval which contains [3,6].
- It is a set which contains [3,6].
- It is the interval [3,6].
- It is a closed interval.
- It is the interval [2,4].
- It is a set which contains 3 and 6.
- It is a bounded set.
- Let be a continuous function such that and . The most we can say about the set is that
- It is a set which contains [3,6].
- PARTIALLY. This is true, but more can be said about f.
- It is a bounded set which contains [3,6].
- INCORRECT. f need not be bounded.
- It is the interval [3,6].
- It is an open interval which contains [3,6].
- INCORRECT. The set need not be open.
- It is an interval which contains [3,6].
- It is a bounded interval which contains [3,6].
- INCORRECT. f need not be bounded.
- It is a set with contains both 3 and 6.
- INCORRECT. Much more can be said about f!
- Let be a uniformly continuous function such that and . The most we can say about the set is that
- It is a set which contains [3,6].
- It is a bounded set which contains [3,6].
- PARTIALLY. This is correct, but more can be said about f.
- It is an interval which contains [3,6].
- PARTIALLY. This is correct, but more can be said about f.
- It is an open bounded interval which contains [3,6].
- INCORRECT. The set need not be open.
- It is a bounded interval which contains [3,6].
- It is an open interval which contains [3,6].
- INCORRECT. The set need not be open.
- It is a bounded set.
Score:
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Quiz: Continuity
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Comments (9)
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.
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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.
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