Piecewise Functions and Taxes (Part 2)

In an another FAQ, I showed how to construct a piecewise function from a tax table on Arizona’s Form 140 Income Tax Form. Other states have descriptions of how income taxes are collected. In 2008, the following description was used to calculate state income taxes in Alabama.

  • For the first $500 in taxable income, the tax rate on that income is 2%.
  • For the next $2500 in taxable income, the tax rate on that income is 4%.
  • On all additional income, the tax rate on that income is 5%.

Let’s use this information to construct a piecewise function T(x) for the income tax as a function of the taxable income x.

Continue reading “Piecewise Functions and Taxes (Part 2)”

How Do You Prove That A Function Is Continuous At Some Point?

Health insurance, taxes and many consumer applications result in a models that are piecewise functions.  To prove these functions are continuous at some point, such as the locations where the pieces meet, we need to apply the definition of continuity at a point.

A function f is continuous at a point x = a if each of the three conditions below are met:

i.  f (a) is defined

ii.  $latex \displaystyle \underset{x\to a}{\mathop{\lim }},f(x)$ is defined

iii.  $latex \displaystyle \underset{x\to a}{\mathop{\lim }},f(x)=f(a)$

In the problem below, we ‘ll develop a piecewise function and then prove it is continuous at two points.

Problem A company transports a freight container according to the schedule below.

  • First 200 miles is $4.00 per mile
  • Next 300 miles is $3.00 per mile
  • All miles over 500 is $2.50 per mile

Let C(x) denote the cost to move a freight container x miles.

a. Find a piecewise function for C(x).

For this function, there are three pieces. The first piece corresponds to the first 200 miles. The second piece corresponds to 200 to 500 miles, The third piece corresponds to miles over 500.

The board below show the function.


Let’s break this down a bit. In the first section, each mile costs $4.50 so x miles would cost 4.5x.

In the second piece, the first 200 miles costs 4.5(200) = 900. All miles over 200 cost 3(x-200). This gives the sum in the second piece.

In the third piece, we need $900 for the first 200 miles and 3(300) = 900 for the next 300 miles. In addition, miles over 500 cost 2.5(x-500).

b. Prove that C(x) is continuous over its domain.

Each piece is linear so we know that the individual pieces are continuous. However, are the pieces continuous at x = 200 and x = 500?

Let’s look at each one sided limit at x = 200 and the value of the function at x = 200.

m212_cont_2_bSince these are all equal, the two pieces must connect and the function is continuous at x = 200. At x = 500,

m212_cont_2_cso the function is also continuous at x = 500.

This means that the function is continuous for x > 0 since each piece is continuous and the  function is continuous at the edges of each piece.

How Do You Construct a Piecewise Function and Prove it is Continuous?

The goal of this problem was to construct a peicewise linear function for individual taxes in Arizona as a function of taxable income. From the Arizona Department of Revenue webpage, I gave the class this information:

Students organized themselves into group to attack several steps in solving this problem. We started this example by simply calculating the tax due on several taxable income levels like $8000, $45,000, and $150,000. My instructions were to show the work like you might want to do on a research poster. Here are some examples of what two groups did.


Both groups did good work and it is easy to follow what they did. However, they get a different amount of tax due. Which value is correct, $5771 or 6221? Why did they get different numbers?

Next I asked the groups to come up with a piecewise linear function that models the information in the table. Here is what several of the groups came up with.

Again, each of the groups did good work. Some are more legible and they will want to focus on communicating their work in the research poster. Finally, I asked them to prove this function is continuous at various taxable income levels. The levels I picked were on the borders of each tax bracket. Points in each bracket are definitely continuous since the pieces are linear. Here is the work each group showed.

Continuous at $50,000?

Continuous at $150,000?

Continuous at $25,000?

There is a lot of good work here. However, has each group proved their point? What is the bar that is set when you are asked to prove a function is continuous at a point?

And here is a big picture question…does it matter that this function is continuous at some point? What are the ramifications of the tax function being discontinuous?

Here is an example I worked out: Piecewise Linear Tax Functions

How Do You Prove a Tax Model is Continuous?

No one likes to pay taxes. It is a fact of life that each of us pay taxes to provide for government services like roads, parks, water service, ect. However, many people have a flawed vision of how the tax they pay changes as their taxable incomes increases.

For instance, one common notion is that the tax you pay jumps when you move to a new tax bracket. In effect, this notion indicates that a small increase in income (like $1) could lead to a huge increase in tax. An example of a function like this is shown below.

In this graph, if your taxable income increases from $10,000 to $10,001, the tax changes from $100 to $150. Yikes!

This graph is an example of a discontinuous function. The discontinuity, at x = 10,000, indicates that this function may not be drawn without lifting a writing device to jump the gap in the graph.

Luckily, tax functions do not behave this way. A typical tax function might look like the one in the graph below.

This is an example of a continuous function. This function may be drawn without having to lift a writing implement from paper. The slope from 0 to 10,000 is 0.01 indicating that the tax rate is 1%. After 10,000, the graph gets slightly steeper. This slope is 0.015 indicating that the tax rate is slightly higher, 1.5%. Higher tax rates for higher incomes is typical in a graduated tax system.

Piecewise linear functions are typically used to model taxes as a function of taxable income.  For the graduated taxes in the graph above, you might be tempted to write

$latex \displaystyle T\left( x \right)=\left\{ \begin{array}{*{35}{l}}
0.01x \\
0.015x \\
\end{array}\quad \begin{array}{*{35}{l}}
\text{if }0\le x\le 10,000 \\
\text{if }x>10,000 \\
\end{array} \right.$

Which of the graphs below matches this function?

To check the continuity of T(x), we start by computing the two-sided limit $latex \displaystyle \underset{x\to 10,000}{\mathop{\lim }}\,T(x)$. Let’s look at the limit from the left side of 10,000:

$latex \displaystyle \underset{x\to 10,{{000}^{-}}}{\mathop{\lim }}\,T(x)=\underset{x\to 10,{{000}^{-}}}{\mathop{\lim }}\,0.01x=100$

The limit from the right of 10,000:

$latex \displaystyle \underset{x\to 10,{{000}^{+}}}{\mathop{\lim }}\,T(x)=\underset{x\to 10,{{000}^{+}}}{\mathop{\lim }}\,0.015x=150$

Since the one-sided limits at x = 10,000 do not match, the two-sided limit does not exist. Looking at the values of each limit, this matches the right hand graph above.

The continuous tax function has the model

$latex \displaystyle T(x)=\left\{ \begin{array}{*{35}{l}}
0.01x \\
0.015x-50 \\
\end{array} \right.\quad \begin{array}{*{35}{l}}
\text{if }0\le x\le 10,000 \\
\text{if }x>10,000 \\

Now when we compute the one-sided limits, they match. The limit from the left side of 10,000:

$latex \displaystyle \underset{x\to 10,{{000}^{-}}}{\mathop{\lim }}\,T(x)=\underset{x\to 10,{{000}^{-}}}{\mathop{\lim }}\,0.01x=100$

The limit from the right of 10,000:

$latex \displaystyle \underset{x\to 10,{{000}^{+}}}{\mathop{\lim }}\,T(x)=\underset{x\to 10,{{000}^{+}}}{\mathop{\lim }}\,0.015x-50=100$

The value of the function at x = 10000 is

$latex \displaystyle T(10,000)=0.01\left( 10,000 \right)=100$

Since the value of the one-sided limits and the value of the function all match, the function is continuous at x = 10,000. So, the function corresponds to the left graph above. By checking these values, we are checking the definition of continuity at x = 10,000,

$latex \displaystyle \underset{x\to 10,000}{\mathop{\lim }}\,T(x)=T(10,000)$

When checking continuity, we need to check each of these values.