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.

- Identify the picture(s) in which we can see what
*f*is doing for all values of*x*in the window.

I. *y* = 5*x* ^{3}- 6*x*^{2} with -0.4 ≤ *x* ≤ 0.4

II. *y* = 5*x*^{3}- 6*x*^{2} with -0.6 ≤* x* ≤ 0.6

I only

II only

Neither I nor II

Both I and II

Q. Four different graphs of the function *f*(*x*) = 2*x*^{2} + 1 are shown below. Which picture best illustrates the fact that if *x* is within 0.06 of 2, then *f*(*x*) is within 0.5 of *f*(2) = 9?

Q. Which statement is true for the function *f* shown below?

If *x* is within 0.5 of 0.5 then *f*(*x*) is within 0.5 of 1.5.

If *x* is within 0.5 of 2 then *f*(*x*) is within 0.5 of 3.

If *x* is within 0.5 of 3 then *f*(*x*) is within 0.5 of 1.

If *x* is within 0.5 of 3.5 then *f*(*x*) is within 0.5 of 1.25.

Q. We have a function *f*. We want *f*(*x*) to be within 0.5 of *f*(0). For which of the following functions do we have a guarantee that we can restrict *x* to find what we want? (*x* must be allowed to move the same distance from 0 in either direction, and *x* may not just be set equal to *0*.)

III only

II and III

I and III

I, II, and III

Q. If *f*(*x*) = 4 - 2*x* then when |*x*-1| < 0.5 we have a guarantee that |*f*(*x*)-2| < 1. Identify *c*, ε, and δ as commonly used in the definition of continuity.

Q. Let *f*(*x*) = 3*x* + 1. Then *f* is continuous at 1 with *f*(1) = 4. Find the largest value of δ for which if *x* is within δ of 1, then *f*(*x*) is within 0.5 of *4*.

0.16

1

Q. Let Then *f* is continuous at 4 with *f*(4) = 2. Find the largest value of δ for which if *x* is within δ of 4, then *f*(*x*) is within 0.25 of *f*(4).

0.25

0.5

1

2

Q. In the formal definition of continuity at *x* = *c*,

ε describes how close we want *f*(*x*) and *f*(*c*), while δ describes how close *x* and *c* need to be.

δ describes how close we want *f*(*x*) and *f*(*c*), while ε describes how close *x* and *c* need to be.

ε and δ may be used interchangeably.

neither ε nor δ is used at all.

Q. If the function *f* is continuous at *c*, then

for some ε>0 we can find a δ>0 such that if |*x*-*c*| < δ then |*f*(*x*)-*f*(c)| < ε.

for some δ>0 we can find an ε>0 such that if |*x*-*c*| < δ then |*f*(*x*)-*f*(c)| < ε.

for every δ>0 we can find an ε>0 such that if |*x*-*c*| < δ then |*f*(*x*)-*f*(c)| < ε.

for every ε>0 we can find a δ>0 such that if |*x*-*c*| < δ then |*f*(*x*)-*f*(c)| < ε.

Q.

- Let ,
*c*= 0, and ε = 0.1.

Which is the largest value of δ such that if |*x*-*c*| < δ, then |*f*(*x*)-*f*(c)| < ε ?

0.09

0.1