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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Section 1.3 • Continuity 105
2
y
20
10
(3, 9)
2 4
x 2 9
if x 3
x 3
f (x)
9 if x 3
x 2 3 if x 3
Figure 24 f is discontinuous at 3.
IN WORDS Visually, the graph of a
function f with a removable
discontinuity has a hole at (c, lim x→c
f (x)).
The discontinuity is removable because
we can fill in the hole by appropriately
defining f . The function will then be
continuous at x = c.
NEED TO REVIEW? The floor function
is discussed in Section P.2, p. 19.
y
2
2
2
Figure 25 f (x) =x
2
4
x
x
Solution Since f (3) = 9, the function f is defined at 3. To check the second condition,
we investigate the one-sided limits.
Teaching Tip
x 2 − 9
lim f (x) = lim
x→3 −
x→3 − x − 3 = lim (x − 3)(x + 3)
= lim
x→3 − x − 3 x→3−(x + 3) = 6
↑
Divide out x − 3
lim f (x) = lim
x→3 +
x→3 +(x 2 − 3) = 9 − 3 = 6
3
Since lim f (x) = lim f (x), then lim f (x) exists. But, lim f (x) = 6 and f (3) = 9,
x −25x
x→3 −
x→3 +
x→3 x→3 fx ( ) =
2
so the third condition of continuity is not satisfied. The function f is discontinuous
x −5x
at 3. ■
Figure 24 shows the graph of f .
NOW WORK Problem 25 and AP® Practice Problems 5, 8, and 11.
fx
The discontinuity at c = 3 in Example 3 is called a removable discontinuity because
we can redefine f at the number c to equal lim f (x) and make f continuous at c. So, in
x→c fx
Example 3, if f (3) is redefined to be 6, then f would be continuous at 3.
DEFINITION Removable Discontinuity
Let f be a function that is defined everywhere in an open interval containing c, except
fx
possibly at c. The number c is called a removable discontinuity of f if the function
is discontinuous at c but lim f (x) exists. The discontinuity is removed by defining (or
x→c
redefining) the value of f at c to be lim f (x).
x→c fx
NOW WORK Problems 13 and 35 and AP® Practice Problem 4.
Determining Whether a Function Is Continuous
EXAMPLE 4 at a Number
Determine whether the floor function f (x) =x is continuous at 1.
Solution The floor function f (x) =x =the greatest integer ≤ x. The floor function f
is defined at 1 and f (1) = 1. But
lim f (x) = lim
x→1 −
x→1−x =0 and lim f (x) = lim
x→1 +
x→1 +x
=1
So, limx does not exist. Since limx does not exist, f is discontinuous at 1. ■
x→1 x→1
Figure 25 illustrates that the floor function is discontinuous at each integer. Also,
none of the discontinuities of the floor function is removable. Since at each integer the
value of the floor function “jumps” to the next integer, without taking on any intermediate
values, the discontinuity at integer values is called a jump discontinuity.
evaluate fc
NOW WORK Problem 53.
We have defined what it means for a function f to be continuous at a number. Now
we define one-sided continuity at a number.
Teaching Tip
DEFINITION One-Sided Continuity at a Number
Let f be a function defined on the interval (a, c]. Then f is continuous from the left
at the number c if
lim f (x) = f (c)
x→c −
Let f be a function defined on the interval [c, b). Then f is continuous from the right
at the number c if
lim f (x) = f (c)
x→c +
TRM Section 1.3: Worksheet 2
This worksheet contains 5 piecewise functions.
The students are asked to determine if the
function is continuous at various values of each
function.
Continuity can also be taught using
graphical and analytic methods. Students
may be surprised to see that the function
looks like a line.
Notice the algebraic simplification:
3
x −25x
( ) =
2
x −5x
2
xx ( −25)
( ) =
xx ( −5)
( x− 5)( x+
5)
( ) =
( x −5)
( ) = x + 5
The x’s and the x – 5’s were removed from
the equation. The values, 0 and 5, were
not in the domain of the original function.
Since both were removed through algebraic
simplification, they are both removable
discontinuities. This function has no other
points of discontinuity.
Common Error
Remind the students that when they
(), they must substitute the
value c into the original function, not the
simplified version of the original function.
If you present the left–right–center check
for continuity at a point, consider having
the students evaluate f() c first. If f()
c
is undefined, the function cannot be
continuous at that point.
Teaching Tip
If a student is given a function and asked to
identify all points of discontinuity, have the
student begin by identifying the domain of
the function. Points not in the domain of the
function are points of discontinuity.
AP® Exam Tip
Although questions about continuity
seldom appear directly on the multiplechoice
portion of the exam, it is
common to see continuity as part of a
multiple-choice question. For example,
a principle of continuity may appear as
one of the answer choices. Also, in a
problem about continuity, it is common
to be given a piecewise function.
Section 1.3 • Continuity 105
TE_Sullivan_Chapter01_PART I.indd 2
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