# 1.6 Continuity

```1.6 Continuity
CALCULUS 9/17/14
Warm-up
The cost (in dollars) of removing p% of the pollutants from the
25,000π
water in a small lake is given by πΆ =
, 0 ≤ π &lt; 100
100−π
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 − πΆ. Explain your results
π₯→100
Warm-up (1.6 Continuity-day 2)
1) Find the limit.
2) lim π π₯
π₯→−1
lim+
π₯→0
1
2+π₯
−
2π₯
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
π₯
−4
π π₯ =
π₯−2
ο§On what interval is this function continuous?
ο§ “The function has a discontinuity at c”
Removable and nonremovable
discontinuities
o Removable- if π can be made continuous by defining
π π at that point
oNonremovable – when the function cannot be made
continuous at x=c
-Ex. π π₯ =
1
π₯
cannot be redefined at x=0
Continuity on a closed interval
ο§ If π is continuous on the open interval (a,b)
ο§ lim+ π π₯ = π(π)
πππ lim− π π₯ = π(π)
π₯→π
π₯→π
ο§Then π is continuous on the closed interval [a,b]
π π₯ = 3−π₯
• Domain:
•Graph
•Continuous
5−π₯,
π π₯ =
2
π₯ − 1,
−1 ≤ π₯ ≤ 2
2&lt;π₯≤3
ο±Is π(π₯) continuous on a closed interval
ο±Closed endpoints?
ο± Continuous on open interval (a,b)?
π₯ + 2,
π π₯ =
2
14 − π₯ ,
−1 ≤ π₯ &lt; 3
3≤π₯≤5
Greatest integer function
ο§ π₯ = greatest integer less than or equal to x
Ex. 5 p. 66
πΆ = 5000 1 +
π₯−1
10,000
+ 3x
-Sketch the graph and analyze the
discontinuities
```