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Continuity of Functions

Continuity of Functions

Continuity of Functions: Go Ahead and Jump True or False

1. Which of the following statements is true? -> A bounded function on a closed interval [a,b] must be continuous.


2. Let . For which interval can we use the Boundedness Theorem to conclude that f must be bounded on that interval?

-> [1,2]

3. Which of the following graphs shows a function that is both bounded and discontinuous on [a,b]? 

-> No. This function is not bounded on [a,b].


4. A continuous function on a closed interval [a,b]

-> must be bounded on that interval but need not attain a maximum value on that interval.

5. What is the maximum value of f(x) = sin(x) on the interval [π,2π)?

-> 0

6. A continuous function on a closed interval [a,b]

-> may attain its maximum and minimum value an infinite number of times each.

7. Let . For which interval(s) can we use the Extreme Value Theorem to conclude that f must attain a maximum and minimum value on that interval?

-> neither (a) nor (b)

8. If f is continuous on [a,b] and f(b)<M<f(a) then the Intermediate Value Theorem tells us how many values of c exist in (a,b) with f(c) = M.


9. Let the function . On which of the following intervals does the IVT guarantee the existence of a value c with f(c) = 0?

-> (-3,0)

10. Which of the following pictures best illustrates the IVT?

-> Picture (d)



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