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Quiz: Continuity

Page history last edited by Terence Tao 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 Formula be a subset of Formula. What does it mean for Formula to be an adherent point of Formula?
    1. For every Formula, there exists a Formula in Formula such that Formula.
      • CORRECT.
    2. For every Formula, there exists a Formula in Formula such that Formula.
      • INCORRECT. This is what it means for x to be a limit point of Formula, not an adherent point.
    3. There exists Formula and Formula such that Formula.
    4. For every Formula and all Formula, we have Formula.
    5. For every Formula, there exists Formula such that Formula.
    6. There exists Formula such that Formula for all Formula in Formula.
    7. There exists Formula such that Formula for all Formula.
  2. Let Formula be a subset of Formula. What does it mean for Formula to not be an adherent point of Formula?
    1. There exists an Formula such that Formula for all Formula in Formula.
      • CORRECT.
    2. There exists an Formula and Formula such that Formula.
    3. For every Formula there exists Formula  such that Formula.
    4. For every Formula we have Formula for every Formula.
    5. For every Formula there exists an Formula such that Formula.
    6. There exists Formula  such that Formula for every Formula.
    7. There exists an Formula such that Formula whenever Formula.
  3. Let Formula be a subset of Formula. What does it mean for Formula to be a limit point of Formula?
    1. For every Formula, there exists a Formula such that Formula.
      • INCORRECT. This is what it means for Formula to be an adherent point of Formula, not a limit point.
    2. For every Formula, there exists a Formula such that Formula.
      • CORRECT.
    3. There exists Formula and  Formula such that Formula.
    4. For every Formula and all Formula, we have Formula.
    5. For every Formula, there exists Formula such that Formula.
    6. There exists Formula such that Formula for all Formula.
    7. There exists Formula such that Formula for all Formula.
  4. Let Formula be a subset of Formula. What does it mean for Formula to be an adherent point of Formula?
    1. There exists a sequence Formula in Formula which converges to Formula.
      • CORRECT.
    2. There exists a sequence Formula in Formula which converges to Formula.
      • INCORRECT. This is what it means for Formula to be a limit point of Formula, not an adherent point.
    3. Every sequence Formula in Formula which converges, converges to Formula.
    4. Every sequence Formula in Formula converges to Formula.
    5. Every Cauchy sequence Formula in Formula converges to Formula.
    6. There exists a sequence Formula in  which converges to Formula.
    7. Every sequence Formula which converges to Formula, must lie in Formula.
  5. Let Formula be a subset of Formula. What does it mean for Formula to be a limit point of Formula?
    1. There exists a sequence Formula in Formula which converges to Formula.
      • INCORRECT. This is what it means for Formula to be an adherent point of Formula, not a limit point.
    2. There exists a sequence Formula of points in Formula which converges to Formula.
      • CORRECT.
    3. Every sequence Formula in Formula which converges, converges to Formula.
    4. Every sequence Formula in Formula converges to Formula.
    5. Every Cauchy sequence Formula in Formula converges to Formula.
    6. Every sequence Formula which converges to Formula, must lie in Formula.
    7. Every sequence Formula which converges to Formula, must lie in Formula.
  6. Let Formula be a subset of Formula, let Formula be a function, and let Formula be an adherent point of Formula. What does it mean for Formula to equal Formula?
    1. For every Formula, there exists a Formula such that Formulafor all Formula for which Formula.
    2. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
    3. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
      • CORRECT.
    4. For every Formula and Formula, we have Formula for all Formula for which Formula.
    5. For every Formula and every Formula, there exists Formula such that Formula if Formula.
    6. For every Formula, there exists a Formula such that Formula and Formula for some Formula.
    7. For every Formula, there exists a Formula such that Formula and Formula for all Formula.
  7. Let Formula be a subset of Formula, let Formula be a function, and let Formula be an element of Formula. What does it mean for Formula to be continuous at Formula?
    1. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
    2. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
    3. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
      • CORRECT.
    4. For every Formula and Formula, we have Formula for all Formula for which Formula.
    5. For every Formula and every Formula, there exists Formula such that Formula if Formula.
    6. For every Formula, there exists a Formula such that Formula and Formula for some Formula.
    7. For every Formula, there exists a Formula such that Formula and Formula for all Formula.
  8. Let Formula be a subset of Formula, let Formula be a function, and let Formula be an adherent point of Formula. What does it mean forFormula to equal Formula?
    1. For every sequence Formula in Formula which converges to Formula, the sequence Formula converges to Formula.
      • CORRECT.
    2. There exists a sequence Formula in Formula converging to Formula, such that Formula converges to Formula.
    3. Whenever Formula converges to Formula, the sequence Formula, must then converge to Formula.
    4. For every sequence Formula in Formula which converges to Formula, the sequence Formula converges to Formula.
    5. Whenever Formula converges to Formula, the sequence Formula, must lie in Formula and converge to Formula.
    6. Whenever Formula converges to Formula, the sequence Formula, must lie in Formula and converge to Formula.
    7. For every sequence Formula in Formula which converges to L, the sequence Formula converges to Formula.
  9. Let Formula be a subset of Formula, let Formula be a function, and let Formula be an element of Formula. What does it mean for Formula to be continuous at Formula?
    1. For every sequence Formula in Formula which converges to Formula, the sequence Formula converges to Formula.
      • CORRECT.
    2. There exists a sequence Formula in Formula converging to Formula, such that Formula converges to Formula.
    3. Whenever Formula converges to Formula, the sequence Formula, must then converge to Formula.
    4. Every sequence Formula which is convergent, converges to Formula.
    5. Whenever Formula converges to f(x), the sequence Formula, must lie in Formula and converge to Formula.
    6. Whenever Formula converges to Formula, the sequence Formula must lie in Formula and converge to Formula.
    7. For every sequence Formula in Formula, the sequence Formula converges to Formula.
  10. Let Formula be a subset of Formula, let Formula be a function. What does it mean for Formula to be continuous on Formula?
    1. For every epsilon > 0 and Formula in X, there exists a delta > 0 such that Formula for all Formula for which Formula.
      • CORRECT.
    2. For every Formula, there exists a Formula such that Formula for all x, y in Formula for which Formula.
      • INCORRECT. This is what it means for Formula to be uniformly continuous on Formula.
    3. For every Formula, there exists a Formula and x,y in Formula such that which Formula and Formula.
    4. For every Formula and Formula, there exists a Formula such that Formula for all y in X for which |y-x| < epsilon.
    5. For every Formula, there exists an Formula and Formula such that Formula and Formula.
    6. For every Formula and Formula, there exists a Formula such that Formula for all Formula for which Formula.
    7. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
  11. Let Formula be a subset of Formula, let Formula be a function. What does it mean for Formula to be uniformly continuous on Formula?
    1. For every Formula and Formula, there exists a Formula such that which Formula for all Formula for which Formula.
      • INCORRECT. This is what it means for Formula to be continuous on Formula.
    2. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
      • CORRECT.
    3. For every Formula, there exists a Formula and x,y in Formula such that Formula and Formula.
    4. For every Formula and Formula in Formula, there exists a Formula such that Formula for all Formula for which Formula.
    5. For every Formula, there exists an Formula and Formula such that Formula and Formula.
    6. For every Formula and Formula in Formula, there exists a Formula such that Formula for all Formula for which Formula.
    7. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
  12. Let Formula be a continuous function such that Formula and Formula. The most we can say about the set Formula 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 Formula be a continuous function such that Formula and Formula. The most we can say about the set Formula 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 Formula be a uniformly continuous function such that Formula and Formula. The most we can say about the set Formula 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:  

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.
----
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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