As we have studied limits, we have gained the intuition that limits measure “where a function is heading.” That is, if , then as is close to 1, is close to 3. We have seen, though, that this is not necessarily a good indicator of what actually is. This can be problematic; functions can tend to one value but attain another. This section focuses on functions that do not exhibit such behavior.
Let be a function defined on an open interval containing .
is continuous at if .
is continuous on if is continuous at for all values of in . If is continuous on , we say is continuous everywhere.
A useful way to establish whether or not a function is continuous at is to verify the following three things:
exists,
is defined, and
.
If is defined near but is not continuous at , then we say that is discontinuous at or has a discontinuity at . We will discuss three types of discontinuities, as seen in Figure 1.6.1.
Let be defined as shown in Figure 1.6.2. Give the interval(s) on which is continuous. ††margin: Λ
SolutionWe proceed by examining the three criteria for continuity.
The limits exists for all between 0 and 3.
is defined for all between 0 and 3, except for . We know immediately that cannot be continuous at .
The limit for all between 0 and 3, except, of course, for .
We conclude that is continuous at every point of except at . Therefore is continuous on .
The floor function,
, returns the largest integer smaller than or equal to the input . (For example, .) The graph of in Figure 1.6.3 demonstrates why this is often called a “step function.”
Give the intervals on which is continuous.
SolutionWe examine the three criteria for continuity.
The limits do not exist at the jumps from one “step” to the next, which occur at all integer values of . Therefore the limits exist for all except when is an integer.
The function is defined for all values of .
The limit for all values of where the limit exist, since each step consists of just a line.
We conclude that is continuous everywhere except at integer values of . So the intervals on which is continuous are
Our definition of continuity on an interval specifies the interval is an open interval. At endpoints or points of discontinuity we may consider continuity from the right or left.
Right Continuous
Let be defined on a closed interval with left endpoint . We say that is continuous from the right at (or right continuous at ) if
Left Continuous
Let be defined on a closed interval with right endpoint . We say that is continuous from the left at (or left continuous at ) if
We can then extend the definition of continuity to closed intervals by considering the appropriate one-sided limits at the endpoints.
Let be defined on the closed interval for some real numbers . Then is continuous on if:
is continuous on ,
is right continuous at and
is left continuous at .
We can make the appropriate adjustments to talk about continuity on half-open intervals such as or if necessary.
Continuity is inherently tied to the properties of limits. Because of this, the properties of limits found in Theorems 1.3.1 and 1.3.3 apply to continuity as well. We will utilize these properties in the following example.
For each of the following functions, give the domain of the function and the interval(s) on which it is continuous.
SolutionWe examine each in turn.
The domain of is . As it is a rational function, we apply Theorem 1.3.3 to recognize that is continuous on all of its domain.
The domain of is all real numbers, or . Applying Theorem 1.3.4 shows that is continuous everywhere.
The domain of is . Applying Theorem 1.3.4 shows that is continuous on its domain of .
The domain of is . We can define the absolute value function as . Each “piece” of this piecewise defined function is continuous on all of its domain, giving that is continuous on and . We cannot assume this implies that is continuous on ; we need to check that , as is the point where transitions from one “piece” of its definition to the other. It is easy to verify that this is indeed true, hence we conclude that is continuous everywhere.
The following theorem states how continuous functions can be combined to form other continuous functions.
Let and be continuous functions on an interval , let be a real number and let be a positive integer. The following functions are continuous on .
Sums/Differences:
Constant Multiples:
Products:
Quotients: (as long as on )
Powers:
Roots: (if on or is odd)
The proofs of each of the parts of Theorem 1.6.1 follow from the Basic Limit Properties given in Theorem 1.3.1. We will prove the product of two continuous functions is continuous now.
We know that and are continuous at so by definition we have
Therefore,
Let be continuous on , where the range of on is , and let be continuous on . Then
is continuous on .
Now knowing the definition of continuity we can re-read Theorem 1.3.4 as giving a list of functions that are continuous on their domains.
The following functions are continuous on their domains.
()
In the following example, we will show how we apply the previous theorems.
State the interval(s) on which each of the following functions is continuous.
The two square-root terms are continuous on the intervals and , respectively. As is continuous only where each term is continuous, is continuous on the intersection of these two intervals: . A graph of is displayed in Figure 1.6.4.
The functions and are each continuous everywhere, hence their product is, too.
Theorem 1.6.3 states that is continuous “on its domain.” Its domain includes all real numbers except odd multiples of . Thus is continuous on
or, equivalently, on .
The domain of is . The range of is , but if we restrict its domain to its range is . So restricting to the domain of restricts its output is , on which is defined. Thus the domain of is .
A common way of thinking of a continuous function is that “its graph can be sketched without lifting your pencil.” That is, its graph forms a “continuous” curve, without holes, breaks or jumps. While beyond the scope of this text, this pseudo-definition glosses over some of the finer points of continuity. Very strange functions are continuous that one would be hard pressed to actually sketch by hand.
This intuitive notion of continuity does help us understand another important concept as follows. Suppose is defined on and and . If is continuous on (i.e., its graph can be sketched as a continuous curve from to ) then we know intuitively that somewhere on must be equal to , and , and etc. In short, takes on all intermediate values between and . It may take on more values; may actually equal 6 at some time, for instance, but we are guaranteed all values between and 5.
While this notion seems intuitive, it is not trivial to prove and its importance is profound. Therefore the concept is stated in the form of a theorem and illustrated in Figure 1.6.5.
Let be a continuous function on and, without loss of generality, let . Then for every value , where , there exists at least one value in such that
One important application of the Intermediate Value Theorem is root finding. Given a function , we are often interested in finding values of where . These roots may be very difficult to find exactly. Good approximations can be found through successive applications of this theorem. Suppose through direct computation we find that and , where . The Intermediate Value Theorem states that there exists at least one in such that . The theorem does not give us any clue as to where that value is in the interval , just that it exists.
Show that has at least one real root.
SolutionWe must determine an interval on which the function changes from positive to negative values. We start by evaluating at different values. We see that and . As we choose larger positive values of , we can see that values will continue to grow. Looking at negative -values, and so we know must change sign in . Because is a polynomial, it is continuous on all real numbers so is continuous on . By the Intermediate Value Theorem there is a in where . Thus must have at least one real root on .
Note that in the above example you were not asked to find the root, just to show that the function had a root.
There is a technique that produces a good approximation of . Let be the midpoint of the interval and consider . There are three possibilities:
— we got lucky and stumbled on the actual value. We stop as we found a root.
Then we know there is a root of on the interval — we have halved the size of our interval, hence are closer to a good approximation of the root.
Then we know there is a root of on the interval — again,we have halved the size of our interval, hence are closer to a good approximation of the root.
Successively applying this technique is called the Bisection Method of root finding. We continue until the interval is sufficiently small. We demonstrate this in the following example.
Approximate the root of , accurate to three places after the decimal.
SolutionConsider the graph of , shown in Figure 1.6.6. It is clear that the graph crosses the -axis somewhere near . To start the Bisection Method, pick an interval that contains . We choose . Note that all we care about are signs of , not their actual value, so this is all we display.
, , and . So replace with and repeat.
, , and at the midpoint, , we see that . So replace with and repeat. Note that we don’t need to continue to check the endpoints, just the midpoint. Thus we put the rest of the iterations in Figure 1.6.7.
Notice that in the 12 iteration we have the endpoints of the interval each starting with . Thus we have narrowed the zero down to an accuracy of the first three places after the decimal. Using a computer, we have
Either endpoint of the interval gives a good approximation of where is 0. The Intermediate Value Theorem states that the actual zero is still within this interval. While we do not know its exact value, we know it starts with .
This type of exercise is rarely done by hand. Rather, it is simple to program a computer to run such an algorithm and stop when the endpoints differ by a preset small amount. One of the authors did write such a program and found the zero of , accurate to 10 places after the decimal, to be 0.7390851332. While it took a few minutes to write the program, it took less than a thousandth of a second for the program to run the necessary 35 iterations. In less than 8 hundredths of a second, the zero was calculated to 100 decimal places (with less than 200 iterations).
It is a simple matter to extend the Bisection Method to solve problems similar to “Find , where .” For instance, we can find , where . It actually works very well to define a new function where . Then use the Bisection Method to solve .
Similarly, given two functions and , we can use the Bisection Method to solve . Once again, create a new function where and solve .
In Section 4.4 another equation solving method will be introduced, called Newton’s Method. In many cases, Newton’s Method is much faster. It relies on more advanced mathematics, though, so we will wait before introducing it.
This section formally defined what it means to be a continuous function. “Most” functions that we deal with are continuous, so often it feels odd to have to formally define this concept. Regardless, it is important, and forms the basis of the next chapter.
In your own words, describe what it means for a function to be continuous.
In your own words, describe what the Intermediate Value Theorem states.
What is a “root” of a function?
Given functions and on an interval , how can the Bisection Method be used to find a value where ?
T/F: If is defined on an open interval containing , and exists, then is continuous at .
T/F: If is continuous at , then exists.
T/F: If is continuous at , then .
T/F: If is continuous on , then .
T/F: If is continuous on and , then is continuous on .
T/F: The sum of continuous functions is also continuous.
In Exercises 11–18., a graph of a function is given along with a value . Determine if is continuous at ; if it is not, state why it is not.
In Exercises 19–22., determine if is continuous at the indicated values. If not, explain why.
In Exercises 23–36., give the intervals on which the given function is continuous.
Exercises 39–42. test your understanding of the Intermediate Value Theorem.
Let be continuous on where and . Does a value exist such that ? Why/why not?
Let be continuous on where and . Does a value exist such that ? Why/why not?
Let be continuous on where and . Does a value exist such that ? Why/why not?
Let be a function on where and . Does a value exist such that ? Why/why not?
In Exercises 43–46., find the value(s) of and so that the function is continuous on .
In Exercises 47–50., sketch the graph of a function that has the following properties.
is discontinuous at 3, but continuous from the left at 3, and continuous elsewhere.
is discontinuous at -1 and 2, but continuous from the right at -1 and continuous from the left at 2, and continuous elsewhere.
has a jump discontinuity at -2 and an infinite discontinuity at 4 and is continuous elsewhere.
has a removable discontinuity at 2, is continuous only from the left at 5, and is continuous elsewhere.
In Exercises 51–54., show that the functions have at least one real root.
Give an example of function for which does not exist.