2.3 Continuity
GOAL: FIND CONTINUITY AT A POINT. FIND
CONTINUOUS FUNCTIONS. FIND
COMPOSITES. USE THE INTERMEDIATE
VALUE THEOREM (IVT)
Definition of Continuous Functions
 Continuous functions are used to describe how a
body moves through space and how speed of a
chemical reaction changes with time.
 Any function y=f(x)whose graph is continuous when
graph can be sketched without lifting your pencil
 lim 𝑓(𝑥) = 𝑓(𝑐) when x is continuous
𝑥→𝑐
Where is the graph continuous and
discontinuous
a)Where is f(x) continuous
−∞, 2 2, ∞
b)Is it continuous at x=0,
and x=6
Yes (0,1)(6,-4)
c)Where is f(x)discontinuous
lim+ 𝑓(𝑥
0
lim− 𝑓(𝑥
4.84
lim 𝑓(𝑥
DNE
𝑥→2
e)
𝑥→2
𝑥→2
At x=2
d)What kind of discontinuity
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Intermediate Value Theorem (IVT)
 In a continuous graph, lim 𝑓(𝑥) = 𝑓(𝑐
𝑥→𝑐
lim+ 𝑓(𝑥) = 𝑓(𝑎) 𝑎𝑛𝑑 lim− 𝑓(𝑥) = 𝑓(𝑏
𝑥→𝑎
𝑥→𝑏
If f(x) is continuous on that closed interval [a,b]
𝑡ℎ𝑒𝑛 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦
𝑛𝑢𝑚𝑏𝑒𝑟 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑓 𝑎 𝑎𝑛𝑑 𝑓(𝑏)
𝐼𝑓 𝑎 < 𝑐 < 𝑏, 𝑡ℎ𝑒𝑛 𝑓(𝑎) < 𝑓(𝑐) < 𝑓(𝑏)
Show there is a solution
 A number is exactly one less that its cube
 𝑥 = 𝑥3 − 1
0 = 𝑥3 − 𝑥 − 1
𝑓(1) = −1

𝑓(2) = 5
A solution is determined by f(x)=0
Types of discontinuity
Removable
Removable
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Infinite
Identify the discontinuity (where and type)
3
𝑓 𝑥 =
𝑥−1
2
2 − 𝑥 𝑖𝑓 𝑥 < 1
𝑥 2 + 1 𝑖𝑓 𝑥 ≥ 1
2𝑥 2 − 5𝑥 − 42
2𝑥 + 7
Composition
 If the function is continuous then all rules apply for
sum, difference, product, quotient, and multiples.
 Hint let u=substitution
 𝐼𝑠 𝑖𝑡 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠

𝑙𝑒𝑡 𝑢 =
3
;3
𝑥−2
3
3
𝑥−2
𝑢 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢
𝑇ℎ𝑒𝑛: 𝑥 ≠ 2
Homework:
 InterActMath.com 2.3