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 $150. Yikes!
This graph is an example of a discontinuous function. The discontinuity, at x = 10,000, indicates that this function may not be drawn without lifting a writing device to jump the gap in the graph.
Luckily, tax functions do not behave this way. A typical tax function might look like the one in the graph below.
This is an example of a continuous function. This function may be drawn without having to lift a writing implement from paper. The slope from 0 to 10,000 is 0.01 indicating that the tax rate is 1%. After 10,000, the graph gets slightly steeper. This slope is 0.015 indicating that the tax rate is slightly higher, 1.5%. Higher tax rates for higher incomes is typical in a graduated tax system.
Piecewise linear functions are typically used to model taxes as a function of taxable income. For the graduated taxes in the graph above, you might be tempted to write
0.01x \\
0.015x \\
\end{array}\quad \begin{array}{*{35}{l}}
\text{if }0\le x\le 10,000 \\
\text{if }x>10,000 \\
\end{array} \right.$
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
$latex \displaystyle T(x)=\left\{ \begin{array}{*{35}{l}}
0.01x \\
0.015x-50 \\
\end{array} \right.\quad \begin{array}{*{35}{l}}
\text{if }0\le x\le 10,000 \\
\text{if }x>10,000 \\
\end{array}$
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.