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!

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)?

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)