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| Figure 1.4 |
In Figure 1.4, the closed circle represents the final point on the line where the function is defined, whereas, the open circle represents the fact that the function is defined on that line only up to that point, but not including it. Because of the jump discontinuity, the limit of the function as x approaches 1 does not exist. However, the function does have two one-sided limits:
The first equation states that the limit as x approaches 1 from the left (indicated by a superscript minus sign after the 1) of ƒ(x) equals 1. The second equation states that the limit as x approaches 1 from the right (indicated by the superscript plus sign after the 1) of ƒ(x) equals 2.
A formal definition for the left-hand limit might look like this:
Definition 1.2: Left-Hand Limit
Let ƒ(x) be defined on the open interval (α,x0) (where α < x). Then the statement
means that for each ε > 0, there exists a δ > 0 such that
x0 - δ < x < x0 implies that y0 - ε < ƒ(x) < y0.■
A similar definition can be constructed for right-handed limits, where the last line would read:
x0 < x < x0 + δ implies that y0 < ƒ(x) < y0 + ε
It is possible for a one-sided limit not to exist. Such is the case for the function ƒ(x) = sin(1/x), which was seen in Lesson 1.3.
The Existence of a Limit
Theorem 1.1: The Existence of a Limit
The limit limx→x0 ƒ(x) exists if and only if both one-sided limits exist at that point and both one-sided limits are equal to each other. Furthermore, the value of the limit, if it exists, is equal to the value of either one-sided limit.■
If a function ƒ(x) is defined at x0 and its limit exists at x0, it does not follow that limx→x0 ƒ(x) = ƒ(x0). For example, consider the function
The plot of this function is shown below:
I put the tick marks on the frame and omitted the axes so you could see a little clearer what is going on. You can see that there is a jump discontinuity at x = 0, where the function "jumps" from x2 to 1 at x = 0. For this function, ƒ(0) = 1; however, limx→x0 ƒ(x) = 0.
We are now in a position to formally define the concepts of continuity and discontinuity.
Definition 1.3: Continuity
1. A function ƒ(x) is continuous at the point x0 if and only if the function is defined at x0 [i.e., ƒ(x0) is defined], limit of the function exists at that point, and
2. Similar to point 1, a function is continuous from the left (or the right) at x0 if it is defined at x0, the left-sided (or right-sided) limit exists at that point, and the limx→x0- = ƒ(x0) [or limx→x0+ = ƒ(x0)].
3. A function is continuous on an open interval (a, b) if and only if it is continuous at each point in the interval.
4. A function is continuous on a closed interval [a, b] if and only if it is continuous on the open interval (a, b) and it is continuous from the right at a and continuous from the left at b.
4. A function which is continuous on the interval (-∞, ∞) is said to be everywhere continuous.
5. A function that is not continuous at a point or on an interval is said to be discontinuous at that point or on that interval. Furthermore, a function that is discontinuous on the interval (-∞, ∞) is said to be everywhere discontinuous.■
The fourth statement in Definition 1.3 may seem a little funny, but there are indeed function which are everywhere discontinuous. One such function is the Dirichlet function, which is defined as:
Not only is the Dirichlet function everywhere discontinuous, but in fact, the limit of the function does not even exist at any point along the entire number line.
Types of Discontinuities
Discontinuities come in two flavors, removable discontinuities, and non-removable discontinuities. A discontinuity of ƒ(x) at x0 is removable if it is possible to define or redefine ƒ(x0) such that the function becomes continuous at that point.
The discontinuity in Figure 1.5 is removable because ƒ(0) can be redefined as ƒ(0) = 0 to make the function continuous. The discontinuity in Figure 1.4, however, is non-removable because there is no way to redefine ƒ(1) such that the function becomes continuous. Another form of non-removable discontinuity occurs when the function exhibits unbounded behavior. For example, the functions ƒ(x) = 1/x and ƒ(x) = 1/x^2 (Shown below in Figures 1.6 and 1.7) both have non-removable discontinuities at x = 0:
Recall the function ƒ1(x) = (x4 - 1)/(x - 1) from Lesson 1.1, in which we factored the numerator into (x3 + x2 + x + 1)(x - 1). This function (shown to the right in Figure 1.8) had a discontinuity at x = 1, which we removed by cancelling the factor (x - 1) from the numerator and denominator to get ƒ2(x) = x3 + x2 + x + 1, which is the same as ƒ2(x), except it is continuous at the point x = 1, rather than being undefined at that point. In general, when a common factor can be cancelled in the numerator and the denominator, then the discontinuity that resulted from that factor being in the denominator is removable and can be removed by cancelling the factor.
Properties of Continuity
Theorem 1.2 Properties of Continuity
If b is a real number and ƒ and g are continuous at x = c, then the following functions are also continuous at c:
1. bƒ
2. ƒ ± g
3. ƒg
4. ƒ/g, provided g(c) ≠ 0
If g is continuous at c and ƒ is continuous at g(c), then the composite function (ƒ ◦ g)(x) = ƒ(g(x)) is continuous at c.■
Some types of function that are continuous at every point in their domain are polynomials, radical function [ƒ(x) = x1/n, for integer n], and the trigonometric functions sin x, cos x, tan x, cot x, sec x, and csc x.
Theorem 1.3 The Intermediate Value Theorem
If ƒ is continuous on the closed interval [a, b], then ƒ(x) takes on every value between ƒ(a) and ƒ(b) at least once in that interval.■
The Existence of a Limit
Theorem 1.1: The Existence of a Limit
The limit limx→x0 ƒ(x) exists if and only if both one-sided limits exist at that point and both one-sided limits are equal to each other. Furthermore, the value of the limit, if it exists, is equal to the value of either one-sided limit.■
If a function ƒ(x) is defined at x0 and its limit exists at x0, it does not follow that limx→x0 ƒ(x) = ƒ(x0). For example, consider the function
The plot of this function is shown below:
 |
| Figure 1.5 |
I put the tick marks on the frame and omitted the axes so you could see a little clearer what is going on. You can see that there is a jump discontinuity at x = 0, where the function "jumps" from x2 to 1 at x = 0. For this function, ƒ(0) = 1; however, limx→x0 ƒ(x) = 0.
We are now in a position to formally define the concepts of continuity and discontinuity.
Definition 1.3: Continuity
1. A function ƒ(x) is continuous at the point x0 if and only if the function is defined at x0 [i.e., ƒ(x0) is defined], limit of the function exists at that point, and
2. Similar to point 1, a function is continuous from the left (or the right) at x0 if it is defined at x0, the left-sided (or right-sided) limit exists at that point, and the limx→x0- = ƒ(x0) [or limx→x0+ = ƒ(x0)].
3. A function is continuous on an open interval (a, b) if and only if it is continuous at each point in the interval.
4. A function is continuous on a closed interval [a, b] if and only if it is continuous on the open interval (a, b) and it is continuous from the right at a and continuous from the left at b.
4. A function which is continuous on the interval (-∞, ∞) is said to be everywhere continuous.
5. A function that is not continuous at a point or on an interval is said to be discontinuous at that point or on that interval. Furthermore, a function that is discontinuous on the interval (-∞, ∞) is said to be everywhere discontinuous.■
The fourth statement in Definition 1.3 may seem a little funny, but there are indeed function which are everywhere discontinuous. One such function is the Dirichlet function, which is defined as:
Types of Discontinuities
Discontinuities come in two flavors, removable discontinuities, and non-removable discontinuities. A discontinuity of ƒ(x) at x0 is removable if it is possible to define or redefine ƒ(x0) such that the function becomes continuous at that point.
The discontinuity in Figure 1.5 is removable because ƒ(0) can be redefined as ƒ(0) = 0 to make the function continuous. The discontinuity in Figure 1.4, however, is non-removable because there is no way to redefine ƒ(1) such that the function becomes continuous. Another form of non-removable discontinuity occurs when the function exhibits unbounded behavior. For example, the functions ƒ(x) = 1/x and ƒ(x) = 1/x^2 (Shown below in Figures 1.6 and 1.7) both have non-removable discontinuities at x = 0:
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| Figure 1.6 |
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| Figure 1.7 |
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| Figure 1.8 |
Properties of Continuity
Theorem 1.2 Properties of Continuity
If b is a real number and ƒ and g are continuous at x = c, then the following functions are also continuous at c:
1. bƒ
2. ƒ ± g
3. ƒg
4. ƒ/g, provided g(c) ≠ 0
If g is continuous at c and ƒ is continuous at g(c), then the composite function (ƒ ◦ g)(x) = ƒ(g(x)) is continuous at c.■
Some types of function that are continuous at every point in their domain are polynomials, radical function [ƒ(x) = x1/n, for integer n], and the trigonometric functions sin x, cos x, tan x, cot x, sec x, and csc x.
Theorem 1.3 The Intermediate Value Theorem
If ƒ is continuous on the closed interval [a, b], then ƒ(x) takes on every value between ƒ(a) and ƒ(b) at least once in that interval.■
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