Discrete & Continuous Functions: Definition & Examples

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  • 0:04 Function Definitions
  • 0:53 Discrete Functions in Detail
  • 1:42 Continuous Functions in Detail
  • 3:22 An Example
  • 4:35 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After this lesson, you will understand the differences between discrete functions and continuous functions. You'll learn the one criterion that you need to look at to determine whether a function is discrete or not.

Function Definition

In this lesson, we're going to talk about discrete and continuous functions. Before we look at what they are, let's go over some definitions. A discrete function is a function with distinct and separate values. This means that the values of the functions are not connected with each other. For example, a discrete function can equal 1 or 2 but not 1.5. A continuous function, on the other hand, is a function that can take on any number within a certain interval. For example, if at one point, a continuous function is 1 and 2 at another point, then this continuous function will definitely be 1.5 at yet another point. A continuous function always connects all its values while a discrete function has separations. Now, let's look at these two types of functions in detail.

Discrete Functions in Detail

Discrete functions have noticeable points and gaps in their graphs. Just look at this one:

A Discrete Function
discrete and continuous functions

Even though these points line up, they are not connected. Because they are not connected and the points are distinct values, this function is a discrete function.

You can write the above discrete function as an equation set like this:

A Discrete Function
discrete and continuous functions

You can see how this discrete function breaks up the function into distinct parts. For this particular function, it is telling you that at x = 1, the function equals 1. At x = 2, the function equals 2. And at x = 5, the function equals 5.

Discrete functions are used for things that can be counted. For example, the number of televisions or the number of puppies born. The graph of discrete functions is usually a scatter plot with scattered points like the one you just saw.

Continuous Functions in Detail

Continuous functions, on the other hand, connect all the dots, and the function can be any value within a certain interval. Look at this graph of the continuous function y = 3x, for example:

A Continuous Function
discrete and continuous functions

This particular function can take on any value from negative infinity to positive infinity. Some continuous functions specify a certain domain, such as y = 3x for x >= 0. This means the graph starts at x = 0 and continues to the right from there. With this specific domain, this continuous function can take on any values from 0 to positive infinity.

You can write continuous functions without domain restrictions just as they are, such as y = 3x or with domain restrictions such as y = 3x for x >= 0.

When your continuous function is a straight line, it is referred to as a linear function. The graph of the continuous function you just saw is a linear function.

The continuous function f(x) = x^2, though, is not a linear function. It is not a straight line.

Another Continuous Function
discrete and continuous functions

This continuous function gives you values from 0 all the way up to positive infinity. It doesn't have any breaks within this interval.

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