Calculus
2.4: Continuity and Discontinuity
Objectives:
2.4.1 Describe continuity and be able to distinguish a continuous function from one with
discontinuities.
Materials:
Exploration 2-4a (IRB)
Lesson 1
Warm Up
p. 49-50: Quick Review: Q1-Q10
1. What is meant by the derivative of a function?
2. What is meant by the definite integral of a function?
3. Draw a pair of alternate interior angles.
4. What type of function has a graph like the following:
y




x













5. Sketch the graph of y  cos x
6. Factor: x 2  5 x  6
532001
7. Evaluate: 2000
53
8. Evaluate: 5!
9. No calculator! Divide 50 by ½ and add 3.
Solutions:
Q1. Instantaneous rate of change
Q2. Product of x and y, where x varies and y can vary
3.
4. Exponential function
y





x













5.
6. x  6x 1
7. 53
8. 120 9. 103

I. Continuity at a Point and on an Open Interval
A function is continuous if there are no holes, steps, or asymptotes. Basically, this means
that you can draw the function without lifting your pencil. Some of the most interesting
functions with discontinuities are “piecewise” functions.
Graphing a Piecewise Function Manually
Piecewise functions have multiple branches at different sections of the domain of the
function. To manually graph a piecewise function, graph each piece separately and then
erase the sections that are beyond the domain of that section.
Example 1
Manually graph the following piecewise function:
2 x  4, x  0
y
x0
 x  1,
y




x













Query: Is the function in Example 1 continuous?
Solution: No, there is a step discontinuity.
Example 2
Manually graph the following piecewise function:
e x ,
f x    1
x1 ,
x 1
x 1
y





x













Query: Is the function in Example 2 continuous?
Solution: No, the vertical asymptote is a discontinuity.
Graphing a Piecewise Function with a Calculator
2 x  4, x  0
Graph the function from Example 1, y  
,
x0
 x  1,
by entering the following into Y1:  2 x  4 x  0   x  1 x  0
Example
x  4,

Plot the following function on your calculator: f x   x 2  8x  8,
x  2,

x 2
2 x 5
x 5
a. Does f(x) have a limit as x approaches 2?
No, the function has different values if you approach from the left or from the

right.
b. Is f(x) continuous at x = 2?
No, there is a gap.
c. Does f(x) have a limit as x approaches 5?
d. Is f(x) continuous at x = 5?
“Types” of Discontinuities: Holes, Steps, and Asymptotes
There are three basic types of discontinuities: holes, steps, and asymptotes.
Figure 1: Hole
Not defined at x  c
Figure 3: Step
lim f  x   f  c 
x c
Figure 2: Hole #2
lim f  x  does not exist
x c
Figure 4: Asymptote
Not defined at x  c and lim f  x  does not
x c
exist
Definition of Continuity
A function is continuous at c if:
lim f  x   f  c  .
x c
Continuous on an open interval (a, b): A function that is continuous at each point in the
interval.
Everywhere continuous: A function that is continuous on the entire real line  ,  
Removable Discontinuities: discontinuities that can be “removed” by redefining f  c  .
Holes are removable discontinuities.
Nonremovable Discontinuities: discontinuities that cannot be fixed easily.
(1) Steps
(2) Vertical asymptotes
Identifying Discontinuities
The three types of discontinuities are easily identified by the cartoonish graphs found in
the textbook. However, hole and jump discontinuities are invisible on graphing
calculators. Therefore, you must be able to identify the discontinuities algebraically.
1. Zeros in Denominators of Rational Functions: could be removable or
nonremovable discontinuities.
2. Holes in Piecewise Functions: these occur when there is a singular x-value that is
not in the domain of the function.
3. Steps in Piecewise Functions: these occur when the endpoints of adjacent
branches don’t match up.
4. Toolkit Functions: you must be familiar enough with the elementary functions to
be able to identify vertical asymptotes, i.e. tanx  and ln x .
5. Plot with a Calculator: for unfamiliar functions, you may be able to identify
vertical asymptotes and steps by simply graphing the function. However,
 on the graphs of calculators. Also, you may
remember that holes cannot be seen
want to plot the functions in “dot mode” so that vertical asymptotes don’t appear
to be part of the function.
6. TABLE: If you suspect that there is a discontinuity at a particular x-value, check
the table on your calculator. If an x-value has an ERROR, then there is a
discontinuity.
Describing Continuity
On tests and quizzes, you will be asked to “describe the continuity of a function”. The
following are possible responses to this question:
1. State that the function is “continuous everywhere”
2. Identify the type of discontinuity and list the x-value(s) where the discontinuities
exist.
Activity 1
Put students into groups of either 4 or 5. Each student should plot two of the functions in
the following set. Then, the students should take turn explaining the continuity of their
functions. Students should show their calculators to their partners so that everyone
understands the continuity description.
Graph the following functions. State whether you think each of the functions is
continuous for the entire real number line.






a. y  x 2  1
Continuous
1
b. y 
Discontinuity at x = 2.
x2
sin x
c. y 
Appears to be continuous, but there is a discontinuity at x  0 .
x
x2  4
d. y 
Appears to be continuous, but there is a discontinuity at x  0 .
x2
1x , x  0
e. y  
Discontinuity at x = 0.
1, x  0
x 2 x2 ,
x 2
f. y   x2
Discontinuity at x = 2.
x 2
1,
g. y  tanx 
Discontinuties at x   2 , 32 ,...
h. y  cosx 
x5
i. y 
x 11
j. y  x  int sin x 
1
k. p  x  
x
Continuous everywhere
Vertical asymptote at x = 11

Discontinuities at every integer (graph in dot mode)
There is a domain restriction and x = 0 (vertical asymptote).
Nonremovable discontinuity
 x  1, x  0
l. h  x    2
 x  1, x  0
x2 1
m. v  x  
x 1
Continuous on  ,  
Removable discontinuity at x = -1
n. y  sin x
Continuous on  ,  
o. g  x   3x2  x  2
Continuous on  ,  
Query 1
Which trigonometric functions are continuous on the entire number line?
sin  x  and cos  x 
Query 2
Define an interval for which the other trigonometric functions are continuous.
tan( x) :   2 , 2 
sec( x) :   2 , 2 
cot( x) :  0,  
csc( x) :  0,  
Sometimes you will be asked to discuss the continuity of a function for only a finite part
of the number line (as opposed to the entire set of real numbers).
Example 1
Discuss the continuity of the following functions on the given interval:
f  t   3  9  t 2 , [3,3]
Continuous
Homework: Day 1: p. 50-51: 1-29 odd
Exit Ticket
1. (T/F) If f  c  = L, then lim f ( x)  L .
x c
False. There could be a step discontinuity at x  c .
Lesson 2
Warm Up
Sketch 5 separate graphs that fit the following descriptions:
1.
2.
3.
4.
5.
has a value for f(-2) but has no limit as x approaches -2
is continuous at x = 4 and is “smooth” there
has a value for f(2) and a limit as x approaches, but is not continuous at x = 2.
the limit of f(x) as x approaches 5 is -2, and the value for f(5) is also -2.
f(3) = 5, but f(x) has no limit as x approaches 3 and no vertical asymptote there.
II. One-Sided Limits and Continuity on a Closed
Interval
Previously, we said that if a function approached different values from the left and right
at a point x = c, then the limit did not exist there. Now, we will learn about limits that
only consider one direction.
Limit from the right: limit that only considers values greater than x = c.
lim f  x 
xc
Limit from the left: limit that only considers values less than x = c.
lim f  x 
xc
Figure 2: One-sided limits
Example 2
Find the limit g  x   25  x 2 as x approaches 5 from the left.
0
The limit in Example 2 would be written like: lim 25  x 2  0
x 5
Example 3: Step Function
The greatest integer function, or “step” function, has a series of gap discontinuities.
Figure 3: Step function
Find the limit of the greatest integer function as x approaches 0 from the left and right.
Limit from the left: -1
Limit from the right: 0
In cases such as this, we say that the limit does not exist.
***Limits Exist When: lim f  x   L and lim f  x   L
x c
x c
Definition of Continuity on a Closed Interval
A function f is continuous on the closed interval  a, b if it is:
(1) continuous on the open interval  a, b  and
(2) lim f  x   f  a  and lim f  x   f  b 
xa
x b
Condition (2) means that at each endpoint, the value of the function is equal to its limit
from within the interval. In other words, there can be no jumps or gaps at the endpoints.
Continuity on a Closed Interval can include functions such as f  x   x , which is not
defined for x values less than zero. According to the definition, f  x   x is continuous
on the interval 0, 
Example 4
Discuss the continuity of the function on the closed interval.
g  x   25  x2 ,
5,5
Example 5
Discuss the continuity of the function on the closed interval.
3  x, x  0
f  x   1
,  1, 4
3  2 x, x  0
Properties of Continuity
If f  x  and g  x  are continuous at x  c , then the following functions are also
continuous at c.
1. Scalar multiple: b  f  x 
2. Sum and difference: f  x   g  x 
3. Product: f  x   g  x 
4. Quotient:
f  x
g  x
There are certain functions that are always continuous at every point in their domain.
Example 6
Determine the domains of the example functions in Table 1. These functions are
continuous everywhere in these domains.
Table 1: Continuous Functions
Type of
Function
Polynomial
Rational
Example
Domain
f  x   x3  2 x 2  6 x  3
Radical
x
x  3x  2
f  x  9  x
Trigonometric
f  x   sin  x 
f  x 
 ,  
 , 2   2, 1   1,  
2
 ,9
 ,  
Note that these “continuous” functions are not necessarily continuous for all real
numbers.
Composite Functions are continuous if the individual functions are continuous.
Example 7
Discuss the continuity of the composite function f  g  x   .
f  x 
1
;
x
g  x   x 1
f  g  x  
1
x 1
This falls under Property of Continuity #4. The function is continuous over the domain:
1, 
Example 8
Discuss the continuity of the composite function f  g  x   .
f  x   sin  x  ;
g  x   x2
f  g  x    sin  x 2 
This is clearly a continuous trigonometric function. The function is continuous over the
domain:  ,  
Testing for Continuity
Sometimes it is useful to define an interval over which a function is continuous. Consider
the function f  x   tan  x  . It is not continuous for all real numbers, but we can define
intervals for which it is continuous.
The function has vertical asymptotes (non-removable discontinuities) at x  2 , 32 , etc.
Interval for which f  x   tan  x  is continuous:   2 , 2 
2
 32 
 2

2

3
2
Figure 4: f  x   tan  x 
Homework: Day 2: p. 50-52: 11-19, 31-35 odd
Exit Ticket Questions
1. Sketch the graph of any function f(x) such that
lim f  x   1 and lim f  x   0
x3
x3
Is the function continuous at x  3 ? Explain.
No, the function is not continuous at x  3 .
2