1. science
2. mathematics
3. algebra

10.5 Continuous Functions
Question 1: What does a continuous function look like?
Question 2: How is a limit related to a continuous function?
Among citizens, taxes are a contentious issue. Every person has their own opinion
regarding how much they should be taxed or what parts of their income should be
taxed. Above all, people feel that taxes should be fair. Taxes help to fund the various
functions we expect from the federal government.
In the United States and many other countries, the tax rate increases as taxable income
rises. In many states, a similar state income tax exists. In the state of Arizona (2011),
several tax brackets exist for individuals with corresponding tax rates.
Taxable income is over
Taxable income is not over
Tax rate
$0
$10,000
2.59%
$10,000
$25,000
2.88%
$25,000
$50,000
3.36%
$50,000
$150,000
4.24%
$150,000
And over
4.54%
This table reflects the progressive side of the tax. The more you make, the higher the
tax rate is. However, many people get squeamish when they change income brackets. If
your taxable income increased the following year from $9,999 to $10,001, would the
change in tax rates lead to a jump in the amount you paid?
In this section, we’ll examine this question in the context of continuous functions. We’ll
learn how to recognize a continuous function. This will help us to determine if there are
any jumps in the amount a person pays in Arizona when an income increase moves
them from one tax bracket to another.
1
Question 1: What does a continuous function look like?
Let’s compare two different scenarios for the amount of tax paid for taxable incomes
between 0 and 25,000 dollars.
In the scenario on the left, the amount of tax increases suddenly as taxable income
increases above $10,000. A jump in a function like this is called a discontinuity. It is not
possible to graph a function with a discontinuity without lifting your pen off the paper.
The graph on the right is a continuous function. It is possible to graph a continuous
function without lifting a pen off of the graph.
If taxes in Arizona were to behave like the graph on the left, moving up to higher bracket
would cause a jump in tax. However, the graph on the right displays no jump. Both
graphs are slightly steeper above 10,000 because of the higher tax rate in that bracket.
We’ll see in Question 2 that the amount of tax as a function of taxable income is a
continuous function like in the graph on the right.
Functions can be discontinuous for several reasons. As we saw above, a graph can
have a jump in it. Other graphs are not defined at some point leading to a discontinuity.
2
Figure 1 – Each graph is discontinuous at
x  a since the functions are not defined at x  a ..
In Figure 1, both functions are undefined at x  a since the function is not defined there.
The graph on the left has a small gap in it. The graph on the right is not defined since
there is a vertical asymptote in the graph.
Even if the graph is defined, it may still have a discontinuity.
Figure 2 – Each graph is discontinuous at
x  a even though the functions are defined at x  a .
If the point at x  a does not “connect” the pieces of the graph around it, it is not
possible to draw the function without lifting a pen from paper. This leads to the
discontinuity at x  a .
A function that can be drawn without lifting your pen from the paper is called a
continuous function. Graphs of linear and quadratic functions are examples of
continuous functions.
3
Example 1
Discontinuities
For each part, explain why the function is discontinuous at the given
point.
a.
f ( x ) at x  1
Solution Although the graph has a “kink” at x  1 , that is not the reason
it is discontinuous there. The function is discontinuous since the
function is not defined at x  1 .
b.
g ( x ) at x  3
4
Solution This function is defined at x  3 , but the pieces of the graph do
not connect leading to a vertical gap. This makes the function
discontinuous.
c.
h( x ) at x  2
Solution The function is almost connected at x  2 , but the point where
they should connect is located in the wrong place. This means we
would not be able to graph the function without lifting a pen from the
graph. So the function is discontinuous at x  2 .
The idea of drawing a continuous function without lifting pen from paper is not very
precise. In the next question, we’ll incorporate limits into a definition of continuity.
5
Question 2: How is a limit related to a continuous function?
Continuity at a Point
A function f is continuous at x  a if each of the conditions
below are met.
1. f ( a ) is defined.
2. lim f ( x ) exists.
xa
3. lim f ( x)  f ( a )
xa
Let’s apply this definition to prove a function is continuous at some point. The state of
Arizona uses a piecewise to calculate the amount of tax T ( x ) in dollars. For taxable
income from $0 to $25,000,
for 0  x  10000
0.0259 x
T ( x)  
0.0288 x  29 for 10000  x  25000
where x is the taxable income. Each piece of this function is a line and each line is
continuous. However, do the two lines connect at x  10000 or is there a discontinuity at
that point?
To be continuous, all three conditions of the definition must be satisfied at x  10000 .
Condition 1 requires that T 10000  be defined. According to the function definition,
T 10000   0.0259 10000   259 6
This means that an individual with $10,000 in taxable income would pay $259 in tax. So
the function is defined at x  10000 .
Condition 2 requires that the two sided limit exist at x  10000 . We will compute the one
sided limits and make sure they exist.
lim T ( x)  lim  0.0259 x
x 10000
x 10000
 0.0259 10000 
 259
lim  T ( x)  lim
x 10000
x 10000
 0.0288 x  29 

 0.0288 10000   29
 259
The limit from the left and right are both equal to 259 so the two sided limit lim T  x 
x 10,000
exists.
Condition 3 requires that the two sided limit be equal to T 10000  . For Condition 2, we
showed that
lim T  x   259 x 10,000
For Condition 1 we found that T 10000   259 . This means that
lim T  x   T 10000  x 10,000
The function is continuous at x  10000 .
7
Figure 3 – The function T(x) is continuous at x = 10000.
As taxable income increases from the lower tax bracket to the higher tax bracket, the
amount of tax paid increases continuously. This is due to the fact that the higher rate is
only paid on income above 10,000. In effect, an individual in the higher tax bracket pays
2.59% on the first $10,000 in income and 2.88% on income above $10,000 up to
$25,000. There is no big jump in the amount of tax paid.
Example 2
Continuous Function
The Basic Plus Plan is a medical insurance plan offered by a self
insured trust in Northern Arizona. The total annual cost (in dollars) to an
insured person is
if 350  x  1200
 x  672

BP  x   0.4 x  1392 if 1200  x  27900
12552
if x  27900

where x is the amount of medical charges incurred at the point of
treatment.
a. Prove that BP  x  is continuous at x  1200 .
Solution Check each condition in the definition of continuity at a point.
8
Condition 1:
BP 1200   1200  672  1872 is defined.
Condition 2:
lim BP ( x)  lim   x  672 
x 1200
x 1200
 1200  672
 1872
lim BP ( x)  lim   0.4 x  1392 
x 1200
x 1200
 0.4 1200   1392
 1872
The one side limits are equal so the two sided limit exists.
Condition 3: Since the one sided limits are both equal to 1872, the two
sided limit is also equal to 1872. Additionally, BP (1200)  1872 . So
lim BP  x   BP 1200 
x 1200
and the function is continuous at x  1200 .
b.
Prove that BP  x  is continuous at x  27900 .
Solution Check each condition in the definition of continuity at a point.
Condition 1:
BP  27900   0.4  27900   1392  12552 is defined.
Condition 2:
9
lim BP ( x)  lim
x  27900
x  27900
 0.4 x  1392 
 0.4  27900   1392
 12552
lim BP ( x)  lim  12552
x  27900
x  27900
 12552
The one side limits are equal so the two sided limit exists.
Condition 3: Since the one sided limits are both equal to 12552, the two
sided limit is also equal to 12552. Additionally, BP (27900)  12552 . So
lim BP  x   BP  27900 
x  27900
and the function is continuous at x  27900 .
Each of the three pieces in Example 2 is continuous since they are each lines. Since the
function is continuous at x  1200 and x  27900 , the total annual cost increases
continuously as the medical charges increase.
Figure 4 – The function BP(x) from Example 2.
10