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Lesson 2-5
Continuity
Objectives
• Understand and use the definition of
continuity
• Understand and use the Intermediate Value
Theorem
Vocabulary
• Continuity – no gaps in the curve (layman’s definition)
• Discontinuity – a point where the function is not continuous
• Removable discontinuity – a discontinuity that can be removed by
redefining the function at a point also called a point discontinuity
• Infinite discontinuity – a discontinuity because the function
increases or decreases without bound at a point
• Jump discontinuity – a discontinuity because the function jumps
from one value to another
• Continuous from the right at a number a – the limit of f(x) as x
approaches a from the right is f(a)
• Continuous from the left at a number a – the limit of f(x) as x
approaches a from the left is f(a)
• A function is continuous on an interval if it is continuous at every
number in the interval
Continuity
Definition: A function is continuous at a number a if
lim f(x)
=
f(a)
xa
Note: that the definition implicitly requires three things of the function
1. f(a) is defined (i.e., a is in the domain of f)
2. lim f(x) exisits
xa
3. lim f(x) = f(a)
xa
f has a discontinuity at a, if f is not continuous at a. Note the graphs of
the examples of discontinuities below:
Removable
x² - x - 2
f(x) = -------------x-2
Infinite
f(x) =
1/x if x ≠ 0
0 if x = 0
Removable
f(x) =
Jump
x²/x if x ≠ 0
1 if x = 0
f(x) = [[x]]
Continuity Theorems
lim f(x)
=
f(a)
xa
If f and g are continuous at a and c is a constant,
then the following functions are also continuous at a:
1. f + g
2. f – g
3. cf
4. f • g
5. f / g if g(a) ≠ 0
6. Any polynomial is continuous everywhere; that is,
its continuous on (-∞,∞)
7. The following types of functions are continuous at every number in
their domains:
polynomials
rational functions
trigonometric functions
inverse trigonometric functions
exponential functions
logarithmic functions
root functions
Continuity Examples
|x|
1. f(x) = -----x
is f(x) continuous at x = 1?
2. f(x) = x² + 1
is f(x) continuous at x = 0?
|x| / x
for x ≠ 0
3. f(x) =
is f(x) continuous at x = 0?
0
for x = 0
Continuity Examples (cont)
x² - 4
--------x-2
x≠2
4. f(x) =
is f(x) continuous at x = 2?
4
x=2
1/x
x ≤ -1
is f(x) continuous at x = -1?
5. f(x) =
(x-1)/2
x
-1 < x < 1
is f(x) continuous at x = 1?
x≥1
Continuity Examples (cont)
Is f(x) =  x² + 6x + 9
continuous for all real numbers?
Continuity from a Graph
a) At which points is the graph discontinuous?
b) On what intervals is the graph continuous?
Intermediate Value Theorem (IVT)
If f is continuous on [a, b] and if W is a number between f(a) and
f(b), then there is a number c between a and b such that f(c) = W.
IVT is an existence theorem.
This means there can be more
than one value in [a,b] that will
satisfy the theorem!
f(b)
W
f(a)
(like in the graph)
a c
c
b
Example: You stop at a light. Twenty-five seconds down the street you
look at your speedometer and it reads 35 mph. Sometime between the
light and then your speedometer had to read 25 mph!
IVT Example
Show that f(x) = x3 + 2x - 1 has a zero on the interval [0, 1].
f(x) is a polynomial and is continuous everywhere.
f(0) = -1
and
f(1) = 2
Since f(x) is continuous on [0,1], then the IVT applies and there
must be a c between a and b such that f(c) = 0
Summary & Homework
• Summary:
– Formal Definition of a limit requires finding an
ε and δ (margins of errors around a and f(a)
respectively)
• Homework: pg 133-135: 7, 10, 15-18, 35,
40, 43