Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page. Such functions are called continuous. Other functions have points at which a break in the graph occurs, but satisfy this property over intervals contained in their domains. They are continuous on these intervals and are said to have a discontinuity at a point where a break occurs.
We begin our investigation of continuity by exploring what it means for a function to have continuity at a point. Intuitively, a function is continuous at a particular point if there is no break in its graph at that point.
Continuity at a Point
Before we look at a formal definition of what it means for a function to be continuous at a point, let’s consider various functions that fail to meet our intuitive notion of what it means to be continuous at a point. We then create a list of conditions that prevent such failures.
Our first function of interest is shown in (Figure). We see that the graph of has a hole at . In fact, is undefined. At the very least, for to be continuous at , we need the following conditions:i. is defined.
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,ii. exists.
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:iii. .
Now we put our list of conditions together and form a definition of continuity at a point.
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.
Determining Continuity at a Point, Condition 2
Using the definition, determine whether the function . Justify the conclusion.
Let’s begin by trying to calculate ..
Thus, is defined. Next, we calculate . To do this, we must compute and :
Therefore, does not exist. Thus, is not continuous at 3. The graph of is shown in (Figure).
By applying the definition of continuity and previously established theorems concerning the evaluation of limits, we can state the following theorem.
Continuity of Polynomials and Rational Functions
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.
Types of Discontinuities
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.
These three discontinuities are formally defined as follows:
Continuity over an Interval
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 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 .
The proof of the next theorem uses the composite function theorem as well as the continuity of
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.
The Intermediate Value Theorem
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.