1. 1.6 Continuity
Calculus 9/17/14

2. Warm-up
The cost (in dollars) of removing p% of the pollutants from the water in a small lake is given by 𝐶= 25,000𝑝 100−𝑝 , 0≤𝑝<100
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 𝑥→ 100 − 𝐶. Explain your results

3. 1.6 Continuity

5. What are some examples of continuous functions?
Polynomials – continuous at every real number
Rational functions – continuous at every number in its domain

7. 𝑓 𝑥 = 𝑥 2 −4 𝑥−2 On what interval is this function continuous?
𝑓 𝑥 = 𝑥 2 −4 𝑥−2
On what interval is this function continuous?
“The function has a discontinuity at c”

8. Removable and nonremovable discontinuities
Removable- if 𝑓 can be made continuous by defining 𝑓 𝑐 at that point
Nonremovable – when the function cannot be made continuous at x=c
-Ex. 𝑓 𝑥 = 1 𝑥 cannot be redefined at x=0
Use example from slide before for removable and slide 6 graph a for nonremovable

9. Continuity on a closed interval
If 𝑓 is continuous on the open interval (a,b)
lim 𝑥→ 𝑎 + 𝑓 𝑥 =𝑓(𝑎) 𝑎𝑛𝑑 lim 𝑥→ 𝑏 − 𝑓 𝑥 =𝑓(𝑏)
Then 𝑓 is continuous on the closed interval [a,b]

10. 𝑓 𝑥 = 3−𝑥
Domain:
Graph
Continuous

11. 𝑔 𝑥 = 5−𝑥 , −1≤𝑥≤2 𝑥 2 −1, 2<𝑥≤3
Is 𝑔(𝑥) continuous on a closed interval
Closed endpoints?
Continuous on open interval (a,b)?
Limits from all sides

12. 𝑓 𝑥 = 𝑥+2, −1≤𝑥<3 14− 𝑥 2 , 3≤𝑥≤5

14. Greatest integer function
𝑥 = greatest integer less than or equal to x

15. Ex. 5 p. 66
𝐶= 𝑥−1 10, x
-Sketch the graph and analyze the discontinuities