2.5 - Continuity

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2.5 - Continuity
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Definition Continuity
What does it mean for something to be continuous?
JumpDiscontinuity
Removable Discontinuity
Infinite Discontinuity
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Continuity
Using calculus, explain why these functions are
not continuous at x a.
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Definition Continuity
A function is continuous at a number a if
. This means you must show
that If this statement is false, then the
function is not continuous at x a.
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Definition One Sided Continuity
A function f is continuous from the right at a
number a if and f is continuous from the left
at a if
?
?
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).
- Continuous from the left
- Continuous from the right
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Theorem
If f and g are continuous at a and c is a
constant, then the following functions are also
continuous at a

• f g
• f g
• cf
• fg
• f / g if g(a) ? 0

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Theorem

• Any polynomial is continuous everywhere that is,
  it is continuous on ? (-8, 8).
• Any rational function is continuous whenever it
  is defined that is, it is continuous on its
  domain.

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Theorem
Any of the following types of functions are
continuous at every number in their domain

• Polynomials
• Rational Functions
• Root Functions
• Trigonometric Functions
• Inverse Trigonometric Functions

• Exponential Functions
• Logarithmic Functions

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Theorems
If g is continuous at a and f is continuous at
g(a), then the composite function f(g(x)) is
continuous at a.
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The Intermediate Value Theorem
Suppose that f is continuous on the closed
interval a, b and let N 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)
f(c)N
f
f(b)
a
b
c
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Example
Use the Intermediate Value Theorem to show that
there is a root of the given equation in the
specified interval.