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What is a discrete graph?

The domain and range of a discrete function are discrete sets of values (points), rather than an interval. It's actually easier to think about discrete graphs, which are the graphs of discrete functions.

Discrete graphs don't have continuous lines. Instead, discrete graphs look like plotted points.

To reduce it even further:

• Discrete graphs: dots
• Continuous graphs: smooth lines

Most of the time, discrete functions involve data that are counted.

Things like number of people visiting a store, dollars in a bank account, or number of hours worked—stuff that has a limit to the level of accuracy we can use to measure it—are all examples of discrete data.

Continuous functions involve data can be infinitely broken down. These functions can be linear or nonlinear.

Things like height, weight, and other measurements are all examples of continuous data.

Being able to distinguish between discrete functions and continuous functions will help with graphing the functions.

Number of pets in a household, number of days spent traveling, and amount of money spent on birthday gifts are all examples of things that generate discrete data. You can't have half a pet in a household or it gets really weird and you have to question your definition of a "pet"; if you count any day when you did any traveling, you'll always get a whole number of days; and while you can get as specific as cents (fractions of a dollar) when talking about money, you can't go down to fractions of a cent. This means that all of these numbers can only get so specific, and there'll be gaps between them where we can't ever have any data.

On the other hand, age can be measured down to fractions of a second, and weight can get as specific as grams, micrograms, or even fractions of micrograms, so anything measuring exact age and weight will generate continuous data. These are continuous variables.

The graph above is just one smooth line with no gaps or holes, so it's actually a continuous graph. We don't see the characteristic dots of a discrete function's graph!

This graph is a collection of points, so even though those points form a recognizable shape it is considered discrete. The above graph looks kind of like a quadratic function!

This graph is only one point, so you might not think of it as a collection. However, in math, sometimes a group that only holds one point still counts as a group, so this graph is considered to be discrete!

Another thing we can do is look at the coordinates of discrete data. This comes down to reading the axes: the $x$-axis is the first coordinate, and the $y$-axis is the second coordinate. Let's look at an example:

Start at the origin, move four units to the left, and then scan the vertical line $x = -4$ and include any points that are on it. In this case, $(-4, -8)$ is the only one, but there could be a bunch depending on the data set.

Discrete vs. Continuous

While a continuous graph has a $y$ value for every single $x$ value and will always appear as a single line, a discrete graph only contains information for specific points and will appear as a collection of those points. Discrete graphs are also often called scatter plots. The two examples below are discrete graphs:

Notice that while the first graph suggests a pattern, the second does not. Both are still discrete graphs, as they only contain information about specific points. You can clearly tell it's a discrete graph because the points do not make up a line.

A good point to keep in mind is that a line has an infinite amount of points, which makes it, by definition, a continuous function.

(Video) Discrete vs. Continuous Relationships

by Friendly Math 101

Here's a video by Friendly Math 101 going over the differences between continuous and discrete graphs.

$\large{\text{Summary}}$

Discrete means something that cannot be broken down into fractions or decimals. An example would be people; you cannot have half a person. We typically say discrete things can be counted.

Continuous is something that can and must be able to be broken down into fractions and decimals. Continuous situations are said to be measured, we can measure them. An example of this would be height.

When figuring out if a graph is continuous or discrete we see if all the points are connected. If the line is connected between the start and the end, we say the graph is continuous. If the points are not connected it is discrete.

Let's now look at some application examples of discrete and continuous things. As you can see continuous things can be measured. Time, temperature, weight, distance, and height are all measured.

Finally we're going to figure out if some applications are continuous or discrete relationships.