Quiz: Continuity

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:  

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