However, as we see in Figure , this condition alone is insufficient to guarantee continuity at the point. Although is defined, the function has a gap at. In this example, the gap exists because does not exist. We must add another condition for continuity at —namely,. However, as we see in Figure , these two conditions by themselves do not guarantee continuity at a point.
The function in this figure satisfies both of our first two conditions, but is still not continuous at. We must add a third condition to our list:. Now we put our list of conditions together and form a definition of continuity at a point. A function is continuous at a point if and only if the following three conditions are satisfied:.
A function is discontinuous at a point if it fails to be continuous at. The following procedure can be used to analyze the continuity of a function at a point using this definition. The next three examples demonstrate how to apply this definition to determine whether a function is continuous at a given point. These examples illustrate situations in which each of the conditions for continuity in the definition succeed or fail.
Using the definition, determine whether the function is continuous at. Justify the conclusion. We can see that , which is undefined. Therefore, is discontinuous at 2 because is undefined. The graph of is shown in Figure. Thus, is defined. Next, we calculate. To do this, we must compute and :. Therefore, does not exist. Thus, is not continuous at 3. Last, compare and. We see that. Since all three of the conditions in the definition of continuity are satisfied, is continuous at.
If the function is not continuous at 1, indicate the condition for continuity at a point that fails to hold. By applying the definition of continuity and previously established theorems concerning the evaluation of limits, we can state the following theorem. Polynomials and rational functions are continuous at every point in their domains.
Previously, we showed that if and are polynomials, for every polynomial and as long as. Therefore, polynomials and rational functions are continuous on their domains. We now apply Figure to determine the points at which a given rational function is continuous.
For what values of is continuous? The rational function is continuous for every value of except. Use Figure. As we have seen in Figure and Figure , discontinuities take on several different appearances. We classify the types of discontinuities we have seen thus far as removable discontinuities, infinite discontinuities, or jump discontinuities. Intuitively, a removable discontinuity is a discontinuity for which there is a hole in the graph, a jump discontinuity is a noninfinite discontinuity for which the sections of the function do not meet up, and an infinite discontinuity is a discontinuity located at a vertical asymptote.
Figure illustrates the differences in these types of discontinuities. Although these terms provide a handy way of describing three common types of discontinuities, keep in mind that not all discontinuities fit neatly into these categories.
If is discontinuous at , then. In Figure , we showed that is discontinuous at. Classify this discontinuity as removable, jump, or infinite. To classify the discontinuity at 2 we must evaluate :. Since is discontinuous at 2 and exists, has a removable discontinuity at.
Earlier, we showed that is discontinuous at 3 because does not exist. However, since and both exist, we conclude that the function has a jump discontinuity at 3. The function value is undefined. We see that and. For , decide whether is continuous at 1. If is not continuous at 1, classify the discontinuity as removable, jump, or infinite. Follow the steps in Figure. If the function is discontinuous at 1, look at and use the definition to determine the type of discontinuity.
Now that we have explored the concept of continuity at a point, we extend that idea to continuity over an interval. As we develop this idea for different types of intervals, it may be useful to keep in mind the intuitive idea that a function is continuous over an interval if we can use a pencil to trace the function between any two points in the interval without lifting the pencil from the paper.
In preparation for defining continuity on an interval, we begin by looking at the definition of what it means for a function to be continuous from the right at a point and continuous from the left at a point. A function is said to be continuous from the right at if.
A function is said to be continuous from the left at if. A function is continuous over an open interval if it is continuous at every point in the interval. A function is continuous over a closed interval of the form if it is continuous at every point in and is continuous from the right at and is continuous from the left at.
Analogously, a function is continuous over an interval of the form if it is continuous over and is continuous from the left at. Continuity over other types of intervals are defined in a similar fashion.
Requiring that and ensures that we can trace the graph of the function from the point to the point without lifting the pencil. If, for example, , we would need to lift our pencil to jump from to the graph of the rest of the function over.
State the interval s over which the function is continuous. Since is a rational function, it is continuous at every point in its domain. The domain of is the set. Thus, is continuous over each of the intervals , and. From the limit laws, we know that for all values of in. We also know that exists and exists. Therefore, is continuous over the interval. Use Figure as a guide for solving.
The Figure allows us to expand our ability to compute limits. In particular, this theorem ultimately allows us to demonstrate that trigonometric functions are continuous over their domains. If is continuous at and , then. Before we move on to Figure , recall that earlier, in the section on limit laws, we showed. Consequently, we know that is continuous at 0. In Figure we see how to combine this result with the composite function theorem.
The given function is a composite of and. Since and is continuous at 0, we may apply the composite function theorem. Use Figure as a guide. The proof of the next theorem uses the composite function theorem as well as the continuity of and at the point 0 to show that trigonometric functions are continuous over their entire domains.
We begin by demonstrating that is continuous at every real number. To do this, we must show that for all values of. The proof that is continuous at every real number is analogous. Because the remaining trigonometric functions may be expressed in terms of and their continuity follows from the quotient limit law. As you can see, the composite function theorem is invaluable in demonstrating the continuity of trigonometric functions.
As we continue our study of calculus, we revisit this theorem many times. Functions that are continuous over intervals of the form , where and are real numbers, exhibit many useful properties. Throughout our study of calculus, we will encounter many powerful theorems concerning such functions. The first of these theorems is the Intermediate Value Theorem. Let be continuous over a closed, bounded interval.
If is any real number between and , then there is a number in satisfying. See Figure. Show that has at least one zero.
As such, continuity or lack thereof can only be determined in the domain of the function. This is NOT pedantry. There is no "in its domain" qualifier. It is a continuous function, period. In the OP's example, the function is perfectly continuous and even has a continuous extension to the entire real line.
Excerpt from my answer to this question :. The term continuous function is defined with respect to its domain. Therefore it is crucial to specify the domain of a function, if we want to analyse the function with respect to continuity.
Outside of the domain of a function this function is not continuous, since it's not even defined there. Note that when we talk about discontinuities of a one variable function we classify them as either being a removable discontinuity , a jump discontinuity or an essential resp.
Informally: The domain and codomain specify where the function lives and we can't say anything about the function outside of its region of existence. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. If a function is undefined at a point, is it also discontinuous at that point? Ask Question. Asked 6 years, 2 months ago. Active 3 years, 5 months ago.
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Email address: Your name:. Possible Answers:. Correct answer:. Explanation : Factorize the numerator for the function: The removable discontinuity is since this is a term that can be eliminated from the function. Set the removable discontinutity to zero and solve for the location of the hole.
Report an Error. Explanation : Rewrite the function in its factored form. If possible, find the type of discontinuity, if any:. Explanation : By looking at the denominator of , there will be a discontinuity.
Since the common factor is existent, reduce the function. Find the point of discontinuity for the following function:. There is no point of discontinuity for the function. Explanation : Start by factoring the numerator and denominator of the function.
There is no point fo discontinuity for this function. Find a point of discontinuity for the following function:. There are no discontinuities for this function. There are no points of discontinuity for this function. Find a point of discontinuity in the following function:. There is no point of discontinuity for this function. Copyright Notice.
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