Lecture Section 2 – 4A
Continuity
We will start the discussion of continuity with a general overview of the concept and then progress to
a formal definition. A function is continuous if the graph of the function has no breaks in it. Generally
speaking, a function is continuous if you can draw itʼs graph without picking up your pencil.
You can trace the graph of each function below without picking up your pencil.
There are no BREAKS or GAPS in the graph.
The functions below are continuous for all real values of x
{ x ∈ Reals } or (−∞ ,∞)
Example 1!
Example 2!
f (x) = x 2 + 1!
Example 3
f (x) = 3 x + 1 !
y = x−3
(0,
1)
(3,0)
!
!
f (x) = (x − 2) 3 !
y = 3x !
y = (1 / 2) x
!
!
f (x) = sin(x)!
f (x) = cos(x)
y
1
1
π
f (x) =(x + a) n (x + b) m (x − c) k
2π
x
–1
Lecture 2 – 4A Continuity "
−π
2
–1
where n,m,k ∈Z +
y
y
π
2
3π
2
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x
x
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Many functions, however, will have isolated points or intervals where there is a ”break” or “gap in the
graph. These problem points or gaps are called discontinuities. The graph is discontinuous for the x
values where the are gaps or holes or where there are no values defined for x.
There are 4 major types of discontinuities for the functions in this chapter.
Type 1
There is a “gap” in the graph at x = c .
As the values of x get close to c the
“left side” of the graph is not going to the same place as the right side of the graph.
As the values of x approach c from the right of x = c the values of y approach K.
As the values of x approach c from the left the values of y approach L.
K
L
x=c
The function is continuous for x except where x = c {x ∈ℜ | x ≠ c}
Type 2
There is a “hole” in the graph at x = c .
As the values of x get close to c the “left side” of the graph and the right side of x = c
both graphs are both approaching the same place but there is hole at that place.
The point ( c , k) is not at the location where the two graphs meet.
As the values of x approach c from the right and form the left of x = c the values of y approach L.
There is a hole at (c, K) and a point at (c , L )
K
L
x=c
The function is continuous for x except where x = c {x ∈ℜ | x ≠ c}
Type 3
Lecture 2 – 4A Continuity "
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There is a “break” in the graph at x = c where
The function is undefined at x = c
f (c) is undefined
non zero number
f(c) =
0
The function is continuous for x except where x = c {x ∈ℜ | x ≠ c}
Type 4A
Intervals where x is undefined
Endpoints and domain restrictions
Some functions have an interval where the values of x are not defined.
This will occur with many radical expressions, The square root function is the most common.
The graphs of these functions will have intervals where there are no x values defined.
The graph will not exist in these intervals.
There are no domain values for x to the left of x = c
The function is continuous for x where x > c {x ∈ℜ | x > c}
y=
x−c +L
NOTE: If a function f(x) has an “endpoint” at x = c and the x values are defined only for points on
one side of x = c due to domain restrictions then the function is NOT continuous at x = c but is is
continuous for values of x greater than c.
Type 4B
Lecture 2 – 4A Continuity "
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Intervals where x is undefined
Endpoints and domain restrictions
Some functions have an interval where the values of x are not defined.
This will occur with many logarithmic expressions.
The graphs of these functions will have intervals where there are no x values defined.
The graph will not exist in these intervals.
y = ln(x − c)
There are no domain values for x = c or for values of x to the left of x = c
The function is continuous for x where x > c {x ∈ℜ | x > c}
NOTE” A function could be defined only for points on one side of x = c. The function is NOT defined
at x = c and for the values for x on the side of c + c are also. The function is continuous for x where
x > c {x ∈ℜ | x > c}
Lecture 2 – 4A Continuity "
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Continuity at a POINT where x = c
f (x) is continuous at point c if all three of the following conditions are met:
1. f (c) is defined
If you substitute c into for x you get a real number f (c) . There is a point
at
2.
3.
lim
f (x) = L
x → c+
(
c , f (c)
)
and f (c) = L
As the values for x gets close to x = c from the right the values for y
approach a real number L.
lim
f (x) = L
x→c
As the values for x gets close to x = c from the left the values for y
approach a real number L.
lim
lim
f (x) = L = f (c)
+ f (x) =
x→ c
x → c−
Both sides of the graph are approaching the same y value L as the values for x get close to c from
both sides. The value for L in step 2 is equal to f (c). As the values for x get close to c , the left and
right sides of the graph are both approaching the the point
(
c , f (c)
)
For what values is the function continuous?
Example 1!
f (x) =
x
x−3
Example 2!
f (x) =
"
x+2
(x + 2) (x − 7) "
Example 3
x +2
f (x) = 2
x − 9"
Continuous for"
Continuous for"
Continuous for
{x ∈ Reals | x ≠ 3} !
{x ∈ Reals | x ≠ −2} !
{x ∈ Reals | x > ±3}
Example 5!
Example 6
Example 4!
f (x) = ln(x − 4)
f (x) = ln(2x + 3)
f (x) = ln(4 − x)
Continuous for"
2x + 3 > 0
−3
x>
2
"
Continuous for"
4− x>0
− x > −4
x<4
"
Continuous for
{x ∈ Reals | x > 4} !
{x ∈ Reals | x > −3 / 2} !
{x ∈ Reals | x < 4}
x−4>0
x>4
"
Lecture 2 – 4A Continuity "
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Example 7!
f (x) =
Example 8!
x−3
Example 9
f (x) = 3x + 5
f (x) = 3 − x
x − 3≥ 0
x≥3
3x + 5 ≥ 0
x ≥ 5/3
The values for x < 3
are undefined so
x = 3 is an endpoint
The values for x < 5 / 3
are undefined so
x = 5 / 3 is an endpoint
3− x ≥ 0
− x ≥ −3
x ≤ −3
The values for x > −3
are undefined so
x = 3 is an endpoint
"
"
"
Continuous for"
Continuous for"
Continuous for
{x ∈ Reals | x > 3} !
{x ∈ Reals | x > 5 / 3} !
{x ∈ Reals | x < -3}
Example 10!
⎧ x +1
⎪
w(x) = ⎨ 5
⎪2x + 3
⎩
2
!
⎧ x3 +1
⎪
w(x) = ⎨ 2
⎪x 2 + 3
⎩
if x > 2
if x = 2
if x < 2
if x > 1
if x = 1
if x < 1
The lim x → 1+ = 2
f(1) = 2
+
The lim x → 1 = 5
f(1) = 5
Example 11!
"
The lim x → 1− = 4
−
The lim x → 1 = 5
There is a "break" at x = 1
There are no "breaks"
Continuous for"
"
Continuous for"
{x ∈All Reals} !
!
{x ∈ Reals | x ≠ 1} !
Lecture 2 – 4A Continuity "
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