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
This definition can be turned around into the following
fact.
Fact 1
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.
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Example 1 Given
the graph of f(x), shown below,
determine if f(x) is continuous at  , , and .
Solution
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 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.
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From this example we can get a quick “working” definition of
continuity. A function is continuous on
an interval if we can draw the graph from start to finish without ever once
picking up our pencil. The graph in the
last example has only two discontinuities since there are only two places where
we would have to pick up our pencil in sketching it.
In other words, a function is continuous if its graph has no
holes or breaks in it.
For many functions it’s easy to determine where it won’t be
continuous. Functions won’t be
continuous where we have things like division by zero or logarithms of
zero. Let’s take a quick look at an
example of determining where a function is not continuous.
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Example 2 Determine
where the function below is not continuous.
Solution
Rational functions are continuous everywhere except where
we have division by zero. So all that
we need to is determine where the denominator is zero. That’s easy enough to determine by setting
the denominator equal to zero and solving.
So, the function will not be continuous at t=-3 and t=5.
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A nice consequence of continuity is the following fact.
Fact 2
To see a proof of this fact see the Proof of Various Limit Properties
section in the Extras chapter. With this
fact we can now do limits like the following example.
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Example 3 Evaluate
the following limit.
Solution
Since we know that exponentials are continuous everywhere
we can use the fact above.
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Another very nice consequence of continuity is the
Intermediate Value Theorem.
Intermediate Value
Theorem
All the Intermediate Value Theorem is really saying is that
a continuous function will take on all values between f(a) and f(b). Below is a graph of a continuous function
that illustrates the Intermediate Value Theorem.
As we can see from this image if we pick any value, M, that is between the value of f(a) and the value of f(b) and draw a line straight out from
this point the line will hit the graph in at least one point. In other words somewhere between a and b the function will take on the value of M. Also, as the figure shows
the function may take on the value at more than one place.
It’s also important to note that the Intermediate Value
Theorem only says that the function will take on the value of M somewhere between a and b. It doesn’t say just what that value will
be. It only says that it exists.
So, the Intermediate Value Theorem tells us that a function
will take the value of M somewhere
between a and b but it doesn’t tell us where it will take the value nor does it
tell us how many times it will take the value.
There are important idea to remember about the Intermediate Value
Theorem.
A nice use of the Intermediate Value Theorem is to prove the
existence of roots of equations as the following example shows.
Let’s take a look at another example of the Intermediate
Value Theorem.
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Example 5 If
possible, determine if takes the following values in the interval
[0,5].
(a) Does
? [Solution]
(b) Does
? [Solution]
Solution
Okay, so much as the previous example we’re being asked to
determine, if possible, if the function takes on either of the two values
above in the interval [0,5]. First,
let’s notice that this is a continuous function and so we know that we can
use the Intermediate Value Theorem to do this problem.
Now, for each part we will let M be the given value for that part and then we’ll need to show
that M lives between and . If it does then we can use the Intermediate
Value Theorem to prove that the function will take the given value.
So, since we’ll need the two function evaluations for each
part let’s give them here,
Now, let’s take
a look at each part.
(a) Okay, in this case we’ll define and we can see that,
So, by the
Intermediate Value Theorem there must be a number such that
[Return to Problems]
(b) In
this part we’ll define . We now have a problem. In this part M does not live between and . So, what does this mean for us? Does this mean that in [0,5]?
Unfortunately
for us, this doesn’t mean anything. It
is possible that in [0,5], but is it also possible that in [0,5].
The Intermediate Value Theorem will only tell us that c’s will exist. The theorem will NOT tell us that c’s don’t exist.
In this case it
is not possible to determine if in [0,5] using the Intermediate Value
Theorem.
[Return to Problems]
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Okay, as the previous example has shown, the Intermediate
Value Theorem will not always be able to tell us what we want to know. Sometimes we can use it to verify that a
function will take some value in a given interval and in other cases we won’t
be able to use it.
For completeness sake here is the graph of in the interval [0,5].
From this graph we can see that not only does in [0,5] it does so a total of 4 times! Also note that as we verified in the first
part of the previous example in [0,5] and in fact it does so a total of 3
times.
So, remember that the Intermediate Value Theorem will only
verify that a function will take on a given value. It will never exclude a value from being
taken by the function. Also, if we can
use the Intermediate Value Theorem to verify that a function will take on a
value it never tells us how many times the function will take on the value, it only
tells us that it does take the value.