Warm Up
Read pages 78-84 (stop at Quick
Review)
Take notes
We will have a quiz shortly.
Reading Quiz 2.3
1. Draw a graph of a function that is NOT continuous.
2. Name at least 2 types of discontinuities.
Continuity
2.3
Goal
I will be able to determine continuity at a point, on an interval, and of
a function through use of limits as well as understand the connections
to the Intermediate Value Theorem.
Most of the techniques of calculus require that functions be continuous. A function
is continuous if you can draw it in one motion without picking up your pencil.
A function is continuous at a point if the limit is the same as the value of the
function.
This function has discontinuities at x=1 and
x=2.
2
1
It is continuous everywhere else.
1
2
3
4

Definition
• A function is continuous on an interval if it is continuous at every
point in the interval.
• A continuous function is continuous at every point in its domain.
Continuity test:
If f(x) is continuous at x=a, THEN:
All three must be
true!
3. lim 𝑓 𝑥 = 𝑓(𝑎)
𝑥→𝑎
Example
So f(x) is NOT continuous.
3. lim 𝑓 𝑥 ≠ 𝑓(𝑎)
𝑥→𝑎
3 main types of discontinuities
1. Removable Discontinuity
2. Jump/Gap Discontinuity
3. Infinite Discontinuity
Removable Discontinuity
Non-Removable Discontinuities
Jump/Gap Discontinuity
Infinite Discontinuity
Removable Discontinuity
• A discontinuity we can fix.
• Steps to remove a discontinuity:
1. Find discontinuities (issues like 0 in denominator, or negative under
square root)
2. Factor/simplify original function
3. Plug the whole into the answer from step 2.
4. Write a piecewise function. Called an extended function
Example: Removing a discontinuity
x3  1
f  x  2
x 1
has a discontinuity at
.
x 1
Write an extended function that is continuous at
.
x 1
 x  1  x 2  x  1 1  1  1
x3  1
 lim

lim 2
x 1
x 1 x  1
 x  1 x  1
2
 x3  1
 2 , x  1
f  x    x 1
 3 , x 1
 2
3

2
Note: There is another discontinuity at
that can not be removed.
x  1
Example: Find the extended Function
𝑥2 − 4
𝑓 𝑥 = 2
𝑥 + 9𝑥 + 14
𝑓 𝑥 =
𝑥+2 𝑥−2
𝑥+7 𝑥+2
the (x+2) can cancel, meaning x=-2 is a removable.
X=-7 is an infinite (can’t touch it.)
lim
𝑥→−2
𝑥−2
𝑥+7
−2 − 2
4
=−
−2 + 7
5
Plug in -2 for x.
Final answer
•𝑔 𝑥 =
𝑥 2 −4
,
2
𝑥 +9𝑥+14
𝑥≠2
4
− , 𝑥 = −2
5
Bracket*
You try!
𝑥 2 − 4𝑥
𝑓 𝑥 = 3
𝑥 − 64
𝑓 𝑥 =
𝑥 𝑥−4
𝑥 − 4 𝑥 2 + 4𝑥 + 16
1
lim 𝑓(𝑥) =
𝑥→4
12
Properties of continuous functions
• If f and g are continuous at a point x=c, the the following
combinations are also continuous at x=c
1.
2.
3.
4.
f+g
f-g
f∙g
k∙f for any number k.
5.
𝑓
𝑔
6. 𝑓 𝑜 𝑔
if g(c)≠ 0
(composition)
Check example on bottom of page 82.
Intermediate Value Theorem (IVT)
A Function f(x) that is continuous on a closed interval [a,b] takes on
every y value between f(a) and f(b).
Example using IVT
• Show that there must be a value c on the interval 1 < 𝑥 < 3 such that h(x)= -5
If ℎ 𝑥 = 0.1𝑥 4 − 1.3𝑥 3 + 2𝑥 2 + 1
• Check end points on interval:
h(1)=1.8
h(3)=-8
Since h is a polynomial, it is continuous on [1,3]. Since ℎ 3 < −5 < ℎ(1), there
exists a value c on (1,3) such that h(c)= -5 by IVT.