Warm up
Evaluate the limit
x x
2
1 . lim
x  1
x  3x  4
2
x x6
2 . lim
h0
2
3 . lim
x 1
x 2
2
4. lim
x0
1 h  1
h
x
25  x  5
Section 2.4
Continuity
• SWBAT
– Define continuity and its types
Conceptual continuity
2.4 Continuity
•
This implies :
1. f(a) is defined
2. f(x) has a limit as x approaches a
3. This limit is actually equal to f(a) .
Definition (cont’d)
Types of discontinuity
Removable Discontinuity: “A hole in the graph”
(You can algebraically REMOVE the discontinuity)
Types of discontinuity (cont’d)
Infinite discontinuity:
•Where the graph
approaches an
asymptote
•It can not be
algebraically removed
jump discontinuity
the function “jumps”
from one value to
another.
Example
• Where are each of the following functions
discontinuous, and describe the type of discontinuity
x3
1. f  x   2
x  x  12
x  9 x  20
2. f  x  
x4
2
One-Sided Continuity
• Continuity can occur from just one side:
Continuity on an Interval
• So far continuity has been defined to occur (or
not) one point at a time.
• We can also consider continuity over an entire
interval at a time:
• Continuous on an Interval: it is continuous at
every point on that interval.
Polynomials and Rational
Functions
• Write the interval where this function is
3
2
continuous.
x  2x  1
lim
:
x2
5  3x
5 5
(  , ) ( , )
3 3
Types of Continuous Function
• We can prove the following theorem:
• This means that most of the functions
encountered in calculus are continuous wherever
defined.
1. Lim f(x)
x23. Lim f(x)
x-
5. Lim f(x)
x-2+
7. f(2)
2. Lim f(x)
x2+
4. Lim f(x)
x-26. Lim f(x)
x0
8. f(-2)
Assignment 8
• p. 126 1-31 odd
• Quiz tomorrow –
2.1 through 2.4 Continuity