2.5 - Continuity
1
Definition: Continuity
What does it mean for something to be
continuous?
○
●
○
___________
Discontinuity
___________
Discontinuity
___________
Discontinuity
2
Continuity
Using calculus, explain why these
functions are not continuous at x = a.
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●
○
a
a
3
Continuity – Limit Definition
A function is continuous at a number x = a if
_______________ = __________
With the definition of limit, this means you must show that
_______________ = _______________ = __________
4
Definition: One Sided Continuity
A function f is continuous from the right
at a number a if
lim f ( x)  f (a)
x a
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○
|
a
and f is continuous from the left at a if
lim f ( x)  f (a)
x a 
●
○
|
a
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Definition: Continuity On An Interval
A function f is continuous on an interval if it is
continuous at every number in the interval. (If f is
defined on one side of an endpoint of the interval, we
understand continuous at the endpoints to mean
continuous from the right or continuous from the left).
lim f  x   f a  - Continuous from the left
xa
lim f  x   f  a  - Continuous from the right
x a
6
Theorem
If f and g are continuous at a and c is a constant,
then the following functions are also continuous
at a:
1.
2.
3.
4.
5.
f+g
f–g
cf
fg
f / g if g(a)  0
7
Theorem
(a) Any polynomial is continuous
everywhere; that is, it is continuous on
 = (-∞, ∞).
(b) Any rational function is continuous
whenever it is defined; that is, it is
continuous on its domain.
8
Theorem
Any of the following types of functions are
continuous at every number in their domain:
• Polynomials
• Exponential Functions
• Rational Functions
• Logarithmic Functions
• Root Functions
• Trigonometric Functions
• Inverse Trigonometric Functions
9
Theorems
If f is continuous at b and lim g ( x)  b , then
x a
lim f ( g ( x))  f (b) . In other words,
x a
lim f ( g ( x))  f (lim g ( x))
x a
x a
If g is continuous at a and f is continuous at g(a),
then the composite function f(g(x)) is continuous
at a.
10
The Intermediate Value Theorem
Suppose that f is
continuous on the closed
f(a)
interval [a, b] and let N f(c)=N
f(b)
be any number between
f(a) and f(b). Then there
exists a number c in (a, b)
such that f(c) = N.
f
a
c
b
11
Example
Use the Intermediate Value Theorem to show
that there is a root of the given equation in the
specified interval.
x  x  1; (1, 2)
2
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