Most functions of a real variable have domains that are intervals or union of separate intervals. To study continuity properly, we will restrict our discussion to these functions. Since are working on an interval, there are three types of points that we need to consider-interior points, left-end points, and right-end points. So let use define what it means to be continuous at an interior point, but before we do that, let us review the concept of limit. When talking about the concept of limit, the value of x is approaching a specific value. It will never equal it. The limit value will always be the y-value that the function is approaching, but it may not equal it. Now to define the concept of continuity at an interior point.

DEFINITION:

A function f is continuous at an interior point x = c of its domain if

What this is saying is this, when determining if a function is continuous at a specific point, there are three items that must be satisfied. Here are the three items.

1.

The function evaluated at the specific value must exist.

2.

The limit of the function as x approaches that value must exist.

3.

The limit of the function must equal the value of the function at that point.

There are four types of discontinuities, and they are the following.

1.

Removable point of discontinuity. A removable point of discontinuity occurs when you can cancel out the factor that causes the denominator to be zero.

EXAMPLE 1:

SOLUTION:

x = -4 is a removable point of discontinuity.

2.

A jump discontinuity. A jump discontinuity is when the function has a visible break in the graph. The break is like a stair step.

EXAMPLE 2:

Notice that the graph jumps from one value to another at x = -1.

3.

An infinite discontinuity. This occurs at vertical asymptotes where the function goes to ± ¥ as x approaches the value.

EXAMPLE 3:

Notice that this function goes to ± ¥ at the vertical asymptote, x = 0.

4.

Graph oscillates. This is a function that bounces up and down as x approaches a value.

EXAMPLE 4:

As x approaches 0, the graph of f (x) oscillates between 1 and -1.

Now let us talk about continuity at endpoints of an interval.

DEFINITION:

A function f is continuous at a left end-point x = a of its domain if

and continuous at a right end-point x = b of its domain if

FACT:

A function is continuous at an interior point c of its domain if and only if it is both right continuous and left continuous at c.

How do we determine if a function is continuous at a point? There is a test we can use, and if all three of its conditions are met, then the function is continuous at that point. Here is the test.

A function f (x) is continuous at x = c if and only if it meets the following three conditions.

1.

f (c) exists.

2.

3.

EXAMPLE 5:

Given

Answer the following questions.

1.

Does f (-1) exist?

2.

3.

4.

Is f continuous at x = -1?

SOLUTION:

1.

Does f (-1) exist? Yes!

f (-1) = (-1) ² - 1 = 1 - 1 = 0

2.

Yes! This is a right-hand limit, so I will approach -1 from the right-hand side.

3.

Yes!

4.

Is f continuous at x = -1? Yes! The three conditions of the continuity test are satisfied, therefore it is continuous at x = -1.

To determine the points, at which a function is continuous, all we need to do is to determine the domain of the function. Since the points that make the function undefined are not included in the domain, then the only points that are left are where the function is continuous. The points that make the function undefined fit into the four types of discontinuities that we discussed earlier in this set of notes.

EXAMPLE 6:

SOLUTION: Let us find the domain for this function. As you should notice, x ¹ -2, so the domain for this function is (-¥ , -2) È (-2, ¥ ). These are the points where this function is continuous.

EXAMPLE 7:

SOLUTION: Again, let us find the domain for this function. When x = -1, | x | + 1 = 1 + 1 = 2. In fact, the denominator will never be zero, so the domain of this function is (-¥ , ¥ ). Therefore this function is continuous on (-¥ , ¥ ).

EXAMPLE 8:

SOLUTION: When is the cos x = 0? When x = ± (2n + 1)p /2, the cos x = 0. Therefore the domain of this function is

and the function is continuous on this domain.

As you should notice that there is a connection between the domain of a function and where the function is continuous. The function is continuous on the domain of the function and nowhere else. So find out where a function is continuous, find the domain of the function.

Now suppose we are given two functions that are continuous at a point x = c. We can combine these two functions to come up with another function that is continuous at x = c.

FACT:

If the functions f and g are continuous at x = c, then the following functions are continuous at x = c:

1.

f + g and f - g

2.

fg

3.

kf, where k is any constant

4.

f/g, provided g (c) ¹ 0

5.

(f (x)) ^{m/ n}, provided (f (x)) ^{m/ n} is defined on an interval containing c, and m and n are integers.

What is the domain of a polynomial function? You should recall that the domain of a polynomial function is the interval (-¥ , ¥ ). Therefore, all polynomial functions are continuous on the interval (-¥ , ¥ ). What is the domain of a ration function? The domain of a rational function is all the points that do not make the denominator zero (i.e. any points that makes the denominator zero are excluded from the domain). Therefore, the domain of a rational function is where the function is continuous.

FACT:

Sine and cosine are continuous for all values of x. Tangent and secant are continuous for all x such that x ¹ ± (2n + 1)p /2 for n = 0, 1, 2, … Cotangent and cosecant are continuous for all x such that x ¹ ± np where n = 0, 1, 2, …

FACT:

If f is continuous at c, and g is continuous at f (c), then g ° f is continuous at c.

What this fact is saying is that if we can break up a composite function into individual functions, then we can determine the continuity of the composite function by looking at the components.

EXAMPLE 9:

Find the limit of the following composite function.

SOLUTION:

FACT:

A function is called continuous if it is continuous everywhere in its domain.

If a function that is not continuous throughout its entire domain, then it may still be continuous when restricted to particular intervals within its domain.

FACT:

A function f is said to be continuous on an interval I in its domain if

at every interior point c and if the appropriate one-sided limits equal the function values at any endpoints I may contain.

Suppose f (x) is continuous on an interval I and a and b are any two points of I. (See figure 4) Then if y₀ is a number between f (a) and f (b), there exists a number c between a and b such that f (c) = y₀.

This theorem can be used to find the root of a polynomial.

figure 4

EXAMPLE 10:

Find the root of f (x) = x ² - 2x - 15 on the interval [4, 6].

SOLUTION: First of all, I am going to evaluate f at the endpoints of the given interval. If f(a) < 0 < f (b) or f (a) > 0 > f (b), then there is a root on this interval.

f (4) = 16 - 2(4) - 15 = 16 - 8 - 15 = -7

f (6) = 36 - 2(6) -15 = 36 - 12 - 15 = 9

So f (4) < 0 < f (6) and there is a c in [4, 6] such that f (c) = 0. Now we need to find the value of c.

c ² - 2c - 15 = 0 ® (c + 3)(c - 5) = 0 ® c = -3 or c = 5

c = 5 is in the interval [4, 6].