Continuity
Over the last few sections we’ve been using the term “nice enough” to define those
functions that we could evaluate limits by just evaluating the function at the point in
question. It’s now time to formally define what we mean by “nice enough”.
Definition
A function
is said to be
continuous
at
if
A function is said to be continuous on the interval [
a, b
] if it is continuous at each point in the
interval.
This definition can be turned around into the following fact.
Fact 1
If
is continuous at
then,
This is exactly the same fact that we first put down
back
when we started looking at
limits with the exception that we have replaced the phrase “nice enough” with
continuous.
It’s nice to finally know what we mean by “nice enough”, however, the definition
doesn’t really tell us just what it means for a function to be continuous. Let’s take a
look at an example to help us understand just what it means for a function to be
continuous.
Example 1
Given the graph of
f(x)
, shown below, determine if
f(x)
is continuous at
,
,
and
.
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To answer the question for each point we’ll need to get both the limit at that point and the
function value at that point. If they are equal the function is continuous at that point and if they
aren’t equal the function isn’t continuous at that point.
First
.
The function value and the limit aren’t the same and so the function is not continuous at this
point. This kind of discontinuity in a graph is called a
jump discontinuity
. Jump discontinuities
occur where the graph has a break in it is as this graph does.
Now
.
The function is continuous at this point since the function and limit have the same value.
Finally
.
The function is not continuous at this point. This kind of discontinuity is called a
removable
discontinuity
. Removable discontinuities are those where there is a hole in the graph as there is
in this case.
From this example we can get a quick “working” definition of continuity. A function
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 Fall '08
 sc
 Continuity, Intermediate Value Theorem, Limits, Continuous function

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