# Continuity of Functions: Is It Discontinued? Quiz

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

**Continuity of Functions**- 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

*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?

*f*shown below?

*x*is within 0.5 of 0.5 then

*f*(

*x*) is within 0.5 of 1.5.

*x*is within 0.5 of 2 then

*f*(

*x*) is within 0.5 of 3.

*x*is within 0.5 of 3 then

*f*(

*x*) is within 0.5 of 1.

*x*is within 0.5 of 3.5 then

*f*(

*x*) is within 0.5 of 1.25.

*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*.)

*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

*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.

*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).

*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.

*ε*nor

*δ*is used at all.

*f*is continuous at

*c*, then

*ε*> 0 we can find a

*δ*> 0 such that if |

*x*–

*c*| <

*δ*then |

*f*(

*x*) –

*f*(

*c*)| <

*ε*.

*δ*> 0 we can find an

*ε*> 0 such that if |

*x*–

*c*| <

*δ*then |

*f*(

*x*)

*– f*(

*c*)| <

*ε*.

*δ*> 0 we can find an

*ε*> 0 such that if |

*x*–

*c*| <

*δ*then |

*f*(

*x*) –

*f*(c)| <

*ε*.

*ε*> 0 we can find a

*δ*> 0 such that if |

*x*–

*c*| <

*δ*then |

*f*(

*x*) –

*f*(

*c*)| <

*ε*.

- 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 *?