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