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# Discrete and Continuous Functions

## Functions without breaks; removable, jump, and infinite discontinuities.

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Discrete and Continuous Functions

Is a bank account balance a continuous function? How about number of pets per household? Gallons of gas in your car? Number of days that students rode bikes to school in a given week?

These are all functions, but they are different types of functions. This Concept is all about learning the difference.

### Discrete and Continuous Functions

Imagine taking a poll to learn the most popular band in school. You interview a large and representative cross-section of students, asking each one how many CD's they have purchased supporting his/her favorite band. You decide that the band with the most CD's sold should be the 'winner'. Wouldn't you be surprised to see numbers like 3.2 or 5.7 on your graph? Who buys .2 or .7 CD's?

Now imagine comparing the CD count to the age of the students to see if some bands are more popular with particular age groups. You add birthdate information to your graph. Would you be surprised to see an average student age of 15.4 or 16.7 years? Of course not, you would probably be a lot more surprised to see an average of exactly 15 or 16 years old.

The difference between these functions is the topic of this lesson. The number of CD's in the first group is a discrete function, since it is very unlikely that someone would purchase a fraction of a CD. The ages in the second count are a continuous function, since people age constantly and particularly at younger ages keep track more accurately than just "years old".

#### Discrete Functions

A discrete function is a function in which the domain and range are each a discrete set of values, rather than an interval in \begin{align*}\mathbb R\end{align*}. Recall from a prior lesson that an interval includes all values between the specified minimum and maximum. If a function is discrete, it does not include all of the values between two given numbers, but rather only specific values in a particular range.

#### Non-Discrete Functions

A non-discrete function is one that is continuous either on its entire domain, or on intervals within its domain. The term continuous refers to a function whose graph has no holes or breaks. (Note that this is not a formal definition. To formally define continuity requires that we use the concept of limit, which we will examine in the next lesson. For now it is sufficient to focus on what the graph looks like.)

### Examples

#### Example 1

Earlier, you were given some examples of different types of functions.

Identify the examples given as either discrete or continuous:

1. A bank account

The balance in a bank account is counted in dollars and cents, any change is countable and quantifiable. This is an example of a discrete function.

1. The number of pets in a household

Discrete function, since one does not generally have a fraction of a pet.

1. Age of students taking this class

Continuous function, there is no limit to the level of accuracy you could apply to the age of each student since time is continuous.

1. Number of days that students rode bikes to school

Discrete, the question suggests a specific count of number of days.

#### Example 2

Identify the function as either continuous or discrete based on the graph:

This graph shows a continuous function, as there are no holes identified on the line, and also no endpoints.

#### Example 3

Identify the function as either continuous or discrete based on the equation:

\begin{align*}y = x^{3} - 3x\end{align*}

The function is continuous, as there is no restriction on the values which may be input for \begin{align*}x\end{align*}.

Note that this is not a straight line, or even a simple curve like y = x2. A graph need not be straight or simple to be continuous.

For Examples 4 and 5, use the following information.

Mark is working at the local fast food restaurant and earns $7.15 per hour. The following table shows the amount of money he earns by working a particular number of hours per week. Hours Worked Money Earned 1$7.15
3 $21.45 7$50.05
12 $85.80 15$107.25

#### Example 4

Does the table above represent data that is "continuous" or "discrete"? Explain your answer. Write an equation that models the data.

The data can be considered continuous because Mark might work any length of time, resulting in any amount of income.

Mark earns $7.15 in each hour he works. His income can be represented by: income = hours X$7.15 or \begin{align*}y = \7.15x\end{align*}.

#### Example 5

Use your equation to predict what Mark’s salary will be if he works 40 hours.

Mark’s income after 40 hours will be \begin{align*}\7.15 \cdot 40 = \286.00\end{align*}

### Review

Identify each of the following variables as being either discrete or continuous.

1. The number of telephone calls received at school in a given week.
2. The weight of a bag of oranges.
3. The length of a piece of rope.
4. Speed of a truck.
5. The number of misdemeanor arrests in a town.
6. Number of flaws in a bolt of fabric.
7. The population of the Bald Eagle.
8. A person's age.
9. Does the graph below represent a Continuous or a Discrete Domain?
10. The equation \begin{align*}f = 0.305m\end{align*} can be used to convert meters into feet. Is the domain of this function discrete or continuous?
11. Your local gardener tells you that your corn plant will grow 1.25” taller each month. It is now 6’ tall. Write a formula that will tell you how tall your plant is at any time in the future. Is there a continuous or a discrete domain?
12. You can buy T-shirts for $12.00, or hats for$15.00. Write an equation showing how much you will spend (y) for any combination of hats (h) and t-shirts (t) that you purchase. Is the domain discrete or continuous?

For questions 13-15, use the following information:

A local neighborhood homeowners association is asking the community residents to participate in a recycling initiative. At the end of each week, each resident is asked submit the number of plastic containers they recycle to the HOA. The data collected was compiled into the following table:

House # Plastic Containers
1 28
2 49
3 35
4 62
5 41
1. Does this table represent data that is "continuous" or "discrete"? Explain your answer
2. Why can you not really write an equation to model the data?
3. Can you predict how many plastic containers the 6th house on the block will recycle the next week?

To see the Review answers, open this PDF file and look for section 1.6.

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Color Highlighted Text Notes

### Vocabulary Language: English

Continuous

Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For a function to be continuous, the function must be continuous at every single point in an unbroken domain.

Continuous Function

A continuous function is a function without breaks or gaps. It contains an infinite, uncountable number of values.

discontinuities

The points of discontinuity for a function are the input values of the function where the function is discontinuous.

Discrete

A relation is said to be discrete if there are a finite number of data points on its graph. Graphs of discrete relations appear as dots.

Discrete Function

A discrete function is a function that has individual and separated values. The members of the domain of a discrete function may be counted individually.

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.

Infinite discontinuities

Infinite discontinuities occur when a function has a vertical asymptote on one or both sides. This will happen when a factor in the denominator of the function is zero.

Infinite discontinuity

Infinite discontinuities occur when a function has a vertical asymptote on one or both sides. This will happen when a factor in the denominator of the function is zero.

interval

An interval is a specific and limited part of a function.

Jump discontinuities

Inverse functions are functions that 'undo' each other. Formally $f(x)$ and $g(x)$ are inverse functions if $f(g(x)) = g(f(x)) = x$.

Removable discontinuities

Removable discontinuities are also known as holes. They occur when factors can be algebraically canceled from rational functions.

Removable discontinuity

Removable discontinuities are also known as holes. They occur when factors can be algebraically canceled from rational functions.