# Precalculus Assignment - Continuous & Discontinuous Functions

Instructor: John Hamilton

John has tutored algebra and SAT Prep and has a B.A. degree with a major in psychology and a minor in mathematics from Christopher Newport University.

The following homeschool assignment deals with functions, specifically as to whether they are continuous or discontinuous. It also explains the three types of discontinuities. The assignment is designed with the 12th grade student in mind. Updated: 04/09/2021

## Assignment Explanation and Topic Overview

As your students progress from precalculus to calculus, they will want to firmly grasp the difference between continuous functions and discontinuous functions.

In addition to what we mentioned above, this lesson includes an explanation of the intermediate value theorem.

By the conclusion of this lesson, your students will have completed two steps, solved two problems, and delivered two relevant presentations.

Note: The answers to the problems along with explanations are located at the bottom of the page.

## Key Terms

• Continuous function: one whose graph is represented by a single unbroken curve
• Discontinuous function: one whose graph has at least one break in its curve
• Region of continuity: a portion of a given function which has an unbroken curve

### Materials

• Graph paper
• Internet access
• Paper
• Pencil
• Ruler

### Time / Length

• One day to complete the material in this assignment
• One week to turn in two appropriate presentations

## Assignment Instructions for Students

### Step One

#### Part I

Let's get ready to calculate! Okay, it sounds much cooler when they say ''let's get ready to rumble'' in pro wrestling, but at least I'm trying to liven up precalculus.

Anyhow, to start with, let's differentiate between continuous functions and discontinuous functions.

A continuous function is one in which you could take your pencil and trace it on your graph paper, and never would you have to lift your pencil from your page. That's not too difficult to comprehend, right?

On the other hand, a discontinuous function is one in which you would have to lift your pencil at some point from your page.

Problem #1:

Given:

f(x) = (x2 + 1) / (x - 2)

Determine:

If this function is continuous or discontinuous

#### Part II

Here we go! Let's identify those three types of discontinuities. They are:

• Asymptotic discontinuity: involves a graph which approaches a given point but never actually touches the point
• Jump discontinuity: entails a graph which stops at a certain point and then picks up at a new point
• Point or removable discontinuity: features a ''hole'' where some point would normally be located along the graph of a function

#### Part III

Next, what does it mean when we say there might be regions of continuity for functions? This is a pretty straightforward concept. Well, let's say we have a discontinuous function. You already know from earlier in this assignment it means at some point you have to pick up your pencil when drawing the graph of the function.

However, you still have portions of the function which you can draw while keeping your pencil touching the page. Those portions of your graph are known as regions of continuity. Does that make sense? Good!

### Step Two

Now let's apply the intermediate value theorem, shall we?

''Hooray!'' The intermediate value theorem. Um, what's that, teacher?''

Well, this is going to sound a bit abstract, but the theorem states:

Given:

A function f(x)

Which is:

Continuous over an a to b interval

Then:

The function must include every value between f(a) and f(b) over that particular interval

Yeah, I know. Let's solve an actual problem to help you see the IVT a bit more clearly.

Problem # 2:

Utilize the intermediate value theorem to conclude if the function:

f(x) = x2 + 4x + 1

passes through y = 0 on the open interval (a, b), which in this case is (-3, 3)

## Deliverable

Now it's time to show your command of the above precalculus material by designing two relevant presentations.

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