# Continuity of Functions: Go Ahead and Jump Quiz

Think you’ve got your head wrapped around

*? Put your knowledge to the test. Good luck — the Stickman is counting on you!***Continuity of Functions**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*]?

(a) and (b)

(b) and (d)

(c) and (d)

(a), (b), and (d)

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 [0, 1] nor [-1, 0]

both [0, 1] and [-1, 0]

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 . 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)