1.6 Continuity
CALCULUS 9/17/14
Warm-up
The cost (in dollars) of removing p% of the pollutants from the
25,000𝑝
water in a small lake is given by 𝐶 =
, 0 ≤ 𝑝 < 100
100−𝑝
Where C is the cost and p is the percent of pollutants.
A) find the cost of removing 50% of the pollutants
B) What percent of the pollutants can be removed for $100,000?
C) Evaluate
lim − 𝐶. Explain your results
𝑥→100
Warm-up (1.6 Continuity-day 2)
1) Find the limit.
2) lim 𝑓 𝑥
𝑥→−1
lim+
𝑥→0
1
2+𝑥
−
2𝑥
1
2
1.6 Continuity
What are some examples of continuous
functions?
• Polynomials – continuous at every real number
• Rational functions – continuous at every number in its
domain
2
𝑥
−4
𝑓 𝑥 =
𝑥−2
On what interval is this function continuous?
 “The function has a discontinuity at c”
Removable and nonremovable
discontinuities
o Removable- if 𝑓 can be made continuous by defining
𝑓 𝑐 at that point
oNonremovable – when the function cannot be made
continuous at x=c
-Ex. 𝑓 𝑥 =
1
𝑥
cannot be redefined at x=0
Continuity on a closed interval
 If 𝑓 is continuous on the open interval (a,b)
 lim+ 𝑓 𝑥 = 𝑓(𝑎)
𝑎𝑛𝑑 lim− 𝑓 𝑥 = 𝑓(𝑏)
𝑥→𝑎
𝑥→𝑏
Then 𝑓 is continuous on the closed interval [a,b]
𝑓 𝑥 = 3−𝑥
• Domain:
•Graph
•Continuous
5−𝑥,
𝑔 𝑥 =
2
𝑥 − 1,
−1 ≤ 𝑥 ≤ 2
2<𝑥≤3
Is 𝑔(𝑥) continuous on a closed interval
Closed endpoints?
 Continuous on open interval (a,b)?
𝑥 + 2,
𝑓 𝑥 =
2
14 − 𝑥 ,
−1 ≤ 𝑥 < 3
3≤𝑥≤5
Greatest integer function
 𝑥 = greatest integer less than or equal to x
Ex. 5 p. 66
𝐶 = 5000 1 +
𝑥−1
10,000
+ 3x
-Sketch the graph and analyze the
discontinuities