How Is Continuity Related to Limits?

Section 10.5 was all about continuous functions.It is typical for students to be able to “guess” their way through the homework. This is because the idea behind continuous function is visual…does the function have any gaps? However, when you have to prove something is continuous…confusion ensues.

Let’s start by refreshing your memory with what it means for a function to be continuous at a point.

A function f is continuous at a point x = a if each of the three conditions below are met:

      1. (a) is defined
      2. \displaystyle \underset{x\to a}{\mathop{\lim }},f(x) is defined
      3. \displaystyle \underset{x\to a}{\mathop{\lim }},f(x)=f(a)

The first two condition insure that you can actually work out each of the sides in the third condition. The last conditions says the numbers you get when you do this should be equal.

Let’s apply this definition to some of the problems on the homework.

Problem 1 For the function f(x)=\frac{{{x}^{2}}-1}{x+1}.

    1. Find the values where f (x) is discontinuous.
    2. What is the value of the limit as x approaches the discontinuity?
    3. If possible, how could you define the function at the discontinuity to make the function continuous at that point?

Problem 2 For the function f(x)=\frac{{x}}{{x}^{2}+2x}.

    1. Find the values where f (x) is discontinuous.
    2. What is the value of the limit as x approaches the discontinuity?
    3. If possible, how could you define the function at the discontinuity to make the function continuous at that point?

Problem 3 For the function f(x)=\frac{{x}}{{x}^{2}-3x}.

    1. Find the values where f (x) is discontinuous.
    2. What is the value of the limit as x approaches the discontinuity?
    3. If possible, how could you define the function at the discontinuity to make the function continuous at that point?

Problem 4 The cost to move a mobile home is given by the table below.

Cost per Mile Distance
$4.00 0 < x < 150
$3.00 150 < x < 400
$2.50 x > 400
    1. Find a piecewise linear function for the cost to move a mobile home x miles.
    2. Prove that the function in A is discontinuous.

A key part of proving the function is discontinuous is carrying out the limits to PROVE it. You also need to do limits to prove that a function is continuous.