# Continuity of Functions: Is It Discontinued? 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.

- 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.*c*= 2,

*ε*= 1,

*δ*= 0.5

*c*= 2,

*ε*= 0.5,

*δ*= 1

*c*= 1,

*ε*= 1,

*δ*= 0.5

*c*= 1,

*ε*= 0.5,

*δ*= 1

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*, thenfor 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