#### CHECK OUT SHMOOP'S FREE STUDY TOOLS: Essay Lab | Math Shack | Videos

# Continuity of Functions

# 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