3

CONTINUOUS FUNCTIONS

Continuous motion

A continuous function

The definition of "a function is continuous at a value of x"

A removable discontinuity

A theorem about continuous functions

In any real problem of continuous motion, 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.

Compare Example 1 and Problem 2 of Lesson 2.

We are about to see that that is the definition of a function being "continuous at the value c."  But why?

To answer, consider the graph of a function f(x) on the left. That graph is a continuous, unbroken line.  Therefore we want to say that  f(x) is a continuous function -- which it is.  But a function is a relationship between numbers (Topic 3 of Precalculus.).  Any definition of a continuous function therefore must be expressed in terms of numbers. To do that, we must see what it is that makes a graph -- a line -- continuous, and then try to find that same characteristic in numbers.

(To avoid scrolling, the figure above is repeated .)

The graph of f(x) stays connected, for example, at x = c.  The graph of g(x) on the right does not stay connected at x = c.

That is, if we think of each graph having two branches, two parts -- one to the left of x = c, and the other to the right -- then in the continuous graph there is no gap between the two parts.  Those parts share a common boundary, the point (c,  f(c)).  We saw in Lesson 1 that that is what characterizes any continuous quantity.

So that is why the graph of f(x) is continuous at x = c.  How can we mathematically define the sentence, "The function f(x) is continuous at x = c."?

Let us think of the values of f(x) -- those numbers -- being in two parts: one to the left of x = c, and one to the right. Then the limit of f(x) as x approaches c exists.

Why?  Because whether the values of x approach c from the left or from the right, the values of 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 values of f(x) share the common boundary  f(c). Therefore borrowing the word "continuous" from geometry (Definition 1), we say that the function is continuous at x = c.

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.

Here is the definition:

By "every" value, we mean every one that we might name. See The mathematical existence of numbers.

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).

We must apply the definition of "continuous at a value of x," Definition 3. 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 on limits, 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.

Definition 3.

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 15 and 16 of Trigonometry.

Problem 2.   Evaluate

sin 0 = 0

Problem 3.   Evaluate

Problem 4.   Velocity, v(t), is a continuous function of time t.  Let

v(t) = 2t² + 1.

If distance is measured in meters, and the function is defined at
t = 5 sec, then explain why

Since v(t) is a continuous function, then according to the definition, we may write the above. For a definition is an "if and only if" sentence, which is to say, it is reversible. See the Commentary on the Definitions in Euclid's Elements.

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.

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,

Equivalently,

Example 3.   Consider this function:

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(x) therefore is continuous at x = 8. (Definition 3.)

In this same way, we could show that the function is continuous at all values of x except x = 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.

(Definition 3.)

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 5.   Consider this function:

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.

(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:

Next Lesson:  The "limit" infinity (∞)

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