We want to determine if the graph of a function rule relating a square's area to its side length is discrete or continuous. Let's start by finding the function rule. The area of a square is calculated by multiplying its two side lengths

s .

A = s \cdot s \Leftrightarrow A = s^{2}

Since the side lengths must be positive, the area of the square is also positive. The side length and the area can be any positive real number.

We see that the above graph has no holes. Therefore, the function is continuous.

Alternative Solution

How to Know This Without Doing Any Math

We can also think about this without really considering the actual function. When functions are discrete, it is usually because some sort of input would cause the function to not make sense.

Example Discrete Function

Let's think about buying pizzas. We can only buy whole pizzas, or maybe slices. Either way, there are specific values that make sense. The total cost

C

for buying

p

pizzas would be a discrete function.

C as a function of p

For this function, the possible values for

p

are

1,

2,

3,

\dots

Therefore, its graph is formed by isolated points.

Function Relating the Area of a Square to Its Side Length

A square can be any size depending on its side length. This means that the area

A

of a square depends on its side length

s .

A as a function of s

Considering a positive length, the possible values for

s

can be any positive real number, like

1.258383,

or

24.57 .

We can calculate the area the same way as usual. Therefore, its graph is unbroken. Therefore, the function is continuous.

Exercise