3
CONTINUOUS FUNCTIONS
The definition of "a function continuous at a value of x"
A theorem about continuous functions
CONTINUOUS MOTION between two points is motion that does not stop between them. (Lesson 1.) Calculus wants to describe that motion mathematically, both the distance traveled and the speed at any given time, particularly when the speed is not constant. Solving that mathematical problem is one of the first applications of calculus.
In any real problem of continuous motion, then, the distance traveled will be represented by a "continuous function" of the time traveled, because we always treat time as continuous. Therefore, we must investigate what we mean by a continuous function.
A continuous function
The graph on the left is the graph of a continuous function; it is one unbroken line. We say that the function f(x) is continuous at every value of x in the interval [a, b]. In particular, f(x) is continuous at the value x = c.
The graph on the right, however, is discontinuous at x = c. If we think of each function as having two "branches" -- one to the left of x = c, and the other to the right -- then in the continuous function there is no gap between the branches. The endpoints of each branch, which are the boundaries or the limits of the branch, coincide at (c, f(c)). But in the graph on the right, the endpoints of each branch do not coincide.
Now we can see that. The question is, How can we express what we see in the technical language of calculus? How can we mathematically define the sentence, "The function f(x) is continuous at x = c."?
(To avoid scrolling, the figure above is repeated .)
We can do so by observing that the limit of f(x) as x approaches c exists.
Why does it exist? Because whether x approaches c from the left or from the right, f(x) approaches f(c). Those limits are equal. (Definition 2.2.)
For example, if f(x) were the function y = x², and c = 5, then
That is,
The two branches of f(x) share the common boundary, the common limit, the common point (c, f(c)). Therefore that function is continuous at x = c.
(Compare Definition 1.)
In the function g(x), however, the limit of g(x) as x approaches c does not exist. If the left-hand limit were the value g(c), the right-hand limit would not be the same. That function is discontinuous at x = c.
The following then is the definition of a function being continuous at a specific value of x.
DEFINITION 3. A function continuous at a value of x.
We say that a function f(x) that is defined at the value x = c
is continuous at that value
if and only if
as x approaches c as a limit,
f(x) approaches f(c) as a limit. (Definition 2.2.)
In symbols:
if and only
if f(x) is continuous at x = c.
If a function is continuous at every value in an interval, then we say that the function is continuous in that interval. And if a function is continuous in any interval, then we simply call it a continuous function.
Although there are exceptions, calculus is essentially about functions that are continuous at every value in their domains. Prime examples of continuous functions are polynomials (Lesson 2).
Problem 1.
a) Prove that this polynomial,
2x² − 3x + 5,
a) is continuous at x = 1.
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!
We must apply the definition of "continuous at a value of x," Definition 3.1. That is, we must show that when x approaches 1 as a limit, f(x) approaches f(1), which is 4.
And according to the Theorems of Lesson 2, that is true.
f(x) therefore is continuous at x = 1.
b) Can you think of any value of x where that polynomial -- or any
b) polynomial -- would not be continuous?
You should not be able to. Polynomials are continuous everywhere. As x approaches any limit c, any polynomial
P(x) approaches P(c). (Lesson 2)
Problems 4, 5, 6 and 7 of Lesson 2 are examples of functions -- polynomials -- that are continuous at each given value.
In addition to polynomials, the following functions also are continuous at every value in their domains.
Rational functions
Root functions
Trigonometric functions
Inverse trigonometric functions
Logarithmic functions
Exponential functions
These are the functions that one encounters throughout calculus. To evaluate the limit of any one of these as x approaches a value, simply evaluate the function at that value.
| Example 2. Evaluate |  |
| Solution. |  |
The student should have a firm grasp of the basic values of the trigonometric functions. In calculus, they are indispensable. See Topics 16 and 17 of Trigonometry.
| Problem 2. Evaluate |  |
sin 0 = 0
| Problem 3. Evaluate |  |
| * | * | * |
If a function is not continuous at a value, then it is discontinuous at that value. Here is the graph of a function that is discontinuous at x = 0.
| This is the graph of y = | 1 x |
. At x = 0, the function is not defined, |
because division by 0 is an excluded operation. (Skill in Algebra, Lesson 5.) x = 0 is a point of discontinuity. In fact, as x approaches 0 -- whether from the right or from the left -- y does not approach any number
Nevertheless, as x increases continuously in an interval that does not include 0, then y will decrease continuously in that interval. We say,
| "The function y = | 1 x |
is continuous for all values of x except x = 0." |
Equivalently,
| "The function y = | 1 x |
is continuous for all values of x is its domain." |
Example 3. Consider this function:
| f(x) | = | x² − 4 x − 2 |
This function is undefined at x = 2, and therefore it is discontinuous there; however, we will return to this point below.
The function nevertheless is defined at all other values of x, and it is continuous at all other values.
For example, as x approaches 8, then according to the Theorems of Lesson 2, f(x) approaches f(8).
| f(8) = | 60 6 |
= 10. |
f(x) therefore is continuous at x = 8. (Definition 3.1.)
In this same way, we could show that the function is continuous at all values of x except x = 2.
A removable discontinuity
For a function to be continuous at x = c, it must exist at x = c. However, when a function does not exist at x = c, it is sometimes possible to assign a value so that it will be continuous there.
This function
| f(x) | = | x² − 4 x − 2 |
does not exist at x = 2. But for every value of x
2:
| x² − 4 x − 2 |
= | (x + 2)(x − 2) x − 2 |
= | x + 2. |
Therefore, as x approaches 2,
(Compare Example 2 of Lesson 2.) That is,
Now, f(x) is not defined at x = 2 -- but we could define it. We could define it to have the value of that limit We could say,
"At x = 2, let f(x) have the value 4."
If we do that, then f(x) will be continuous at x = 2 -- because the limit at that value will be the value of the function.
When we are able to do that -- define a function at a value where it is undefined and therefore discontinuous -- we say that the function has a removable discontinuity.
f(x) above is such a function.
Problem 2. Consider this function:
| y = | 5x x |
. |
a) For which value of x is this function discontinuous? x = 0.
b) Define the function there so that it will be continuous.
For all values except x = 0, the function has the value 5. Hence, if we define the function to have the value 5 at x = 0, then we have successfully removed the discontinuity.
A theorem about continuous functions
Let f(x) = 5x and g(x) = x3. And let us consider
Since f is continuous, then the limit as x approaches 2 will be the value
f (g(2)).
(See Topic 3 of Precalculus.)
Now let us consider
That is, let us first take the limit of g. But since g is continuous, that limit will be g(2). We will have
In other words:
THEOREM 3.1. If f and g are continuous in an interval containing the value c, then
Next Lesson: The "limit" infinity (∞)
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