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Continuity of Functions: Go Ahead and Jump Quiz

Think you’ve got your head wrapped around Continuity of Functions? Put your knowledge to the test. Good luck — the Stickman is counting on you!
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Q. Which of the following statements is true?


A continuous function on a closed interval [a,b] must be bounded.

A bounded function on a closed interval [a,b] must be continuous.

A continuous function on an open interval (a,b) must be bounded.

A bounded function on an open interval (a,b) must be continuous.

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



[0,1]
[-5,0]
(2,3)
[1,2]
Q. Which of the following graphs shows a function that is both bounded and discontinuous on [a,b]? 



No. This function is continuous on the interval [a,b].
Yes. This function is bounded (in fact, always has the same value) on [a,b], and is discontinuous on [a,b].

No. This function is continuous on [a,b].

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

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



must be bounded on that interval and attain a maximum value on that interval.
must be bounded on that interval but need not attain a maximum value on that interval.
need not be bounded on that interval but must attain a maximum value on that interval.
need not be bounded on that interval and need not attain a maximum value on that interval.
Q. What is the maximum value of f(x) = sin(x) on the interval [π,2π)?



f does not attain a maximum on this interval.
0
1
-1
Q. A continuous function on a closed interval [a,b]



must attain its maximum and minimum value exactly once each.
may attain its maximum and minimum value more than once, but only a finite number of times each.
may attain its maximum and minimum value an infinite number of times each.
need not attain a maximum or minimum value at all.
Q. 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?



[0,1]
[-1,0]
neither (a) nor (b)
both (a) and (b)
Q. 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.

one value of c in (a,b) at which f(c) = M.

all values of c in (a,b) at which f(c) = M.

that there is a value of c in (a,b) with f(c) = M.

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



(-3,-2)
(-3,-1)
(-3,0)
(-3,1)
Q. Which of the following pictures best illustrates the IVT?



Picture (a)

Picture (b)

Picture (c)

Picture (d)

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