VARIABLE AND FUNCTION - FUNCTIONS AND LIMITS - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER VI. FUNCTIONS AND LIMITS

INTRODUCTION

The main body of modern mathematics centers around the concepts of function and limit. In this chapter we shall analyze these notions systematically.

An expression such as

x2 + 2x − 3

has no definite numerical value until the value of x is assigned. We say that the value of this expression is a function of the value of x, and write

x2 + 2x − 3 = f(x).

For example, when x = 2 then 22 + 2.2 − 3 = 5, so that f(2) = 5. In the same way we may find by direct substitution the value of f(x) for any integral, fractional, irrational, or even complex number x.

The number of primes less than n is a function π(n) of the integer n. When a value of n is given, the value π(n) is determined, even though no algebraic expression for computing it is known. The area of a triangle is a function of the lengths of its three sides; it varies as the lengths of the sides vary and is determined when these lengths are given definite values. If a plane is subjected to a projective or a topological transformation, then the coordinates of a point after the transformation depend on, i.e. are functions of, the original coordinates of the point. The concept of function enters whenever quantities are connected by a definite physical relationship. The volume of a gas enclosed in a cylinder is a function of the temperature and of the pressure on the piston. The atmospheric pressure as observed in a balloon is a function of the altitude above sea level. The whole domain of periodic phenomena—the motion of the tides, the vibrations of a plucked string, the emission of light waves from an incandescent filament—is governed by the simple trigonometric functions sin x and cos x.

To Leibniz (1646-1716), who first used the word “function,” and to the mathematicians of the eighteenth century, the idea of a functional relationship was more or less identified with the existence of a simple mathematical formula expressing the exact nature of the relationship. This concept proved too narrow for the requirements of mathematical physics, and the idea of a function, together with the related notion of limit, was subjected to a long process of generalization and clarification, of which we shall give an account in this chapter.

§1. VARIABLE AND FUNCTION

1. Definitions and Examples

Often mathematical objects occur which we are free to choose arbitrarily from a whole set S of objects. Then we call such an object a variable within the range or domain S. It is customary to use letters from the latter portion of the alphabet for variables. Thus if S denotes the set of all integers, the variable X with the domain S denotes an arbitrary integer. We say, “the variable X ranges over the set S,” meaning that we are free to identify the symbol X with any member of the set S. The use of variables is convenient when we wish to make statements involving objects chosen at will from a whole set. For example, if S again denotes the set of integers and X and Y are both variables with the domain S, the statement

X + Y = Y + X

is a convenient symbolic expression of the fact that the sum of any two integers is independent of the order in which they are taken. A particular case is expressed by the equation

2 + 3 = 3 + 2,

involving constants, but to express the general law, valid for all pairs of numbers, symbols having the meaning of variables are needed.

It is by no means necessary that the domain S of a variable X be a set of numbers. For example, S might be the set of all circles in the plane; then X would denote any individual circle. Or S might be the set of all closed polygons in the plane, and X any individual polygon. Nor is it necessary that the domain of a variable contain an infinite number of elements. For example, X might denote any member of the population S of a given city at a given time. Or X might denote any one of the possible remainders when an integer is divided by 5; in this case the domain S would consist of the five numbers 0, 1, 2, 3, 4.

The most important case of a numerical variable—in this case we customarily use a small letter x—is that in which the domain of variability S is an interval axb of the real number axis. We then call x a continuous variable in the interval. The domain of variability of a continuous variable may be extended to infinity. Thus S may be the set of all positive real numbers, x > 0, or even the set of all real numbers without exception. In a similar way we may consider a variable X whose values are the points in a plane or in some given domain of the plane, such as the interior of a rectangle or of a circle. Since each point of the plane is defined by its two coördinates, x, y, with respect to a fixed pair of axes, we often say in this case that we have a pair of continuous variables, x and y.

It may be that with each value of a variable X there is associated a definite value of another variable U. Then U is called a function of X. The way in which U is related to X is expressed by a symbol such as

U = F(X) (read, ”F of X”).

If X ranges over the set S, then the variable U will range over another set, say T. For example, if S is the set of all triangles X in the plane, a function F(X) may be defined by assigning to each triangle X the length, U = F(X), of its perimeter; T will be the set of all positive numbers. Here we note that two different triangles, X1 and X2, may have the same perimeter, so that the equation F(X1) = F(X2) is possible even though X1X2. A projective transformation of one plane, S, onto another, T, assigns to each point X of S a single point U of T according to a definite rule which we may express by the functional symbol U = F(X). In this case F(X1) ≠ F(X2) whenever X1X2, and we say that the mapping of S onto T is biunique (see p. 78).

Functions of a continuous variable are often defined by algebraic expressions. Examples are the functions

image

In the first and last of these expressions, x may range over the whole set of real numbers; while in the second, x may range over the set of real numbers with the exception of 0—the value 0 being excluded since 1/0 is not a number.

The number B(n) of prime factors of n is a function of n, where n ranges over the domain of all natural numbers. More generally, any sequence of numbers, a1, a2, a3, · · ·, may be regarded as the set of values of a function, u = F(n), where the domain of the independent variable n is the set of natural numbers. It is only for brevity that we write an for the nth term of the sequence, instead of the more explicit functional notation F(n) The expressions discussed in Chapter I,

image

are functions of the integral variable n.

If U = F(X) we usually reserve for X the name independent variable, while U is called the dependent variable, since its value depends on the value chosen for X.

It may happen that the same value of U is assigned to all values of X, so that the set T consists of one element only. We then have the special case where the value U of the function does not actually vary; that is, U is constant. We shall include this case under the general concept of function, even though this might seem strange to a beginner, for whom the emphasis naturally seems to lie in the idea that U varies when X does. But it will do no harm—and will in fact be useful—to regard a constant as the special case of a variable whose “domain of variation” consists of a single element only.

The concept of function is of the greatest importance, not only in pure mathematics but also in practical applications. Physical laws are nothing but statements concerning the way in which certain quantities depend on others when some of these are permitted to vary. Thus the pitch of the note emitted by a plucked string depends on the length, weight, and tension of the string, the pressure of the atmosphere depends on the altitude, and the energy of a bullet depends on its mass and velocity. The task of the physicist is to determine the exact or approximate nature of this functional dependence.

The function concept permits an exact mathematical characterization of motion. If a moving particle is concentrated at a point in space with rectangular coordinates x, y, z, and if t measures the time, then the motion of the particle is completely described by giving its coördinates x, y, z as functions of t:

x = f(t), y = g(t), z = h(t).

Thus, if a particle falls freely along the vertical z-axis under the influence of gravity alone,

image

where g is the acceleration due to gravity. If a particle rotates uniformly on a circle of unit radius in the x, y-plane, its motion is characterized by the functions

x = cos ωt, y = sin ωt,

where ω is a constant, the so-called angular velocity of the motion.

A mathematical function is simply a law governing the interdependence of variable quantities. It does not imply the existence of any relationship of “cause and effect” between them. Although in ordinary language the word “function” is often used with the latter connotation, we shall avoid all such philosophical interpretations. For example, Boyle’s law for a gas contained in an enclosure at constant temperature states that the product of the pressure p and the volume v is a constant c (whose value in turn depends on the temperature):

pv = c.

This relation may be solved for either p or v as a function of the other variable,

image

without implying that a change in volume is the “cause” of a change in pressure any more than that the change in pressure is the “cause” of the change in volume It is only the form of the connection between the two variables which is relevant to the mathematician.

Mathematicians and physicists differ sometimes as to the aspect of the function concept on which they put the emphasis. The former usually stresses the law of correspondence, the mathematical operation that is applied to the independent variable x to obtain the value of the dependent variable u. In this sense f () is a symbol for a mathematical operation; the value u = f(x) is the result of applying the operation f() to the number x. On the other hand, the physicist is often more interested in the quantity u as such than in any mathematical procedure by which the values of ucan be computed from those of x. Thus the resistance u of the air to a moving object depends on the velocity v and can be found by experiment, whether or not an explicit mathematical formula for computing u = f(v) is known. It is the actual resistance which primarily interests the physicist and not any particular mathematical formula f(v), except insofar as the study of such a formula may aid in analyzing the behavior of the quantity u. This is the attitude ordinarily taken if one applies mathematics to physics or engineering. In more advanced calculations with functions confusion can sometimes be avoided only by knowing exactly whether one means the operation f() which assigns to x a quantity u = f(x), or the quantity u itself, which may also be considered to depend, in a quite different manner, on some other variable, z. For example, the area of a circle is given by the function u = f(x) = πx2, where x is the radius, and also by the function u = g(z) = z2/4π, where z is the circumference.

Perhaps the simplest types of mathematical functions of one variable are the polynomials, of the form

u = f(x) = a0 + a1x + a2x1 + · · · + anxn,

with constant “coefficients,” a0, a1 · · ·,an. Next come the rational functions, such as

image

which are quotients of polynomials, and the trigonometric functions, cos x, sin x, and tan x = sin x/cos x, which are best defined by reference to the unit circle in the image, η-plane, image2 + η2 = 1. If the point P(image, η) moves on the circumference of this circle, and if x is the directed angle through which the positive image-axis must be rotated in order to coincide with OP, then cos x and sin x are the coördinates of P: cos x = image, sin x = η.

2. Radian Measure of Angles

For all practical purposes angles are measured in units obtained by subdividing a right angle into a number of equal parts. If this number is 90, then the unit is the familiar “degree.” A subdivision into 100 parts would be better adapted to our decimal system, but would represent the same principle of measuring. For theoretical purposes, however, it is advantageous to use an essentially different method of characterizing the size of an angle, the so-called radian measure. Many important formulas involving the trigonometric functions of angles have a simpler form in this system than if the angles are measured in degrees.

To find the radian measure of an angle we describe a circle of radius 1 about the vertex of the angle. The angle will cut out an arc s on the circumference of this circle, and we define the length of this arc as the radian measure of the angle. Since the total circumference of a circle with radius 1 has the length 2π, the full angle of 360° has the radian measure 2π. It follows that if x denotes the radian measure of an angle and y its degree measure, then x and y are connected by the relation y/360 = x/2π or

πy = 180x.

Thus an angle of 90° (y = 90) has the radian measure x = 90π/180 = π/2, etc. On the other hand, an angle of 1 radian (the angle with radian measure x = 1) is the angle that cuts out an arc equal to the radius of the circle; in degrees this will be an angle of y = 180/π = 57.2957 · · · degrees. We must always multiply the radian measure x of an angle by the factor 180/π to obtain its degree measure y.

The radian measure x of an angle is also equal to twice the area A of the sector of the unit circle cut out by the angle; for this area bears to the whole area of the circle the ratio which the arc along the circumference bears to the whole circumference: x/2π = A/π, x = 2A.

Henceforth the angle x will mean the angle whose radian measure is x. An angle of x degrees will be written x°, to avoid ambiguity.

It will become apparent that radian measure is very convenient for analytic operations. For practical use, however, it would be rather inconvenient. Since π is irrational, we shall never return to the same point of the circle if we mark off repeatedly the unit angle, i.e. the angle of radian measure 1. The ordinary measure is so devised that after marking off 1 degree 360 times, or 90 degrees 4 times, we return to the same position.

3. The Graph of a Function. Inverse Functions

The character of a function is often most clearly shown by a simple geometrical graph. If x, u are coördinates in a plane with respect to a pair of perpendicular axes, then linear functions such as

u = ax + b

are represented by straight lines; quadratic functions such as

u = ax2 + bx + c

by parabolas; the function

image

by a hyperbola, etc. By definition, the graph of any function u = J(x) consists of all the points in the plane whose coördinates x, u are in the relationship u = f(x). The functions sin x, cos x, tan x, are represented by the curves in Figures 151 and 152. These graphs show clearly how the values of the functions increase or decrease as x varies.

image

Fig. 151. Graphs of sin x and cos x.

image

Fig. 152. u = tan x.

An important method for introducing new functions is the following. Beginning with a known function, F(X), we may try to solve the equation U = F(X) for X, so that X will appear as a function of U:

X = G(U).

The function G(U) is then called an inverse function of F(X). This process leads to a unique result only if the function U = F(X) defines a biunique mapping of the domain of X onto that of U, i.e. if the inequality X1X2 always implies the inequality F(X1) ≠ F(X2), for only then will there be a uniquely defined X correlated with each U. Our previous example in which X denoted any triangle in the plane and U = F(X) was its perimeter is a case in point. Obviously this mapping of the set S of triangles onto the set T of positive real numbers is not biunique, since there are infinitely many different triangles with the same perimeter. Hence in this case the relation U = F(X) does not serve to define a unique inverse function. On the other hand, the function m = 2n, where n ranges over the set S of integers and m over the set T of even integers, does give a biunique correspondence between the two sets, and the inverse function n = m/2 is uniquely defined. Another example of a biunique mapping is provided by the function

image

Fig. 153. u = x3.

As x ranges over the set of all real numbers, u will likewise range over the set of all real numbers, assuming each value once and only once. The uniquely defined inverse function is

image

In the case of the function

u = x2,

an inverse function is not uniquely determined. For since u = x2 = (−x)2, each positive value of u will have two antecedents. But if, as is customary, we define the symbol image to mean the positive number whose square is u, then the inverse function

image

exists, so long as we restrict x and u to positive values.

The existence of a unique inverse of a function of one variable, u = f(x), can be seen by a glance at the graph of the function. The inverse function will be uniquely defined only if to each value of u there corresponds but one value of x. In terms of the graph, this means that no parallel to the x-axis intersects the graph in more than one point. This will certainly be the case if the function u = f(x) is monotone, i.e. steadily increasing or steadily decreasing as x increases. For example, if u = f(x) is steadily increasing, then for x1 < x2 we always have u1 = f(x1) < u2 = f(x2). Hence for a given value of u there can be at most one x such that u = f(x), and the inverse function will be uniquely defined. The graph of the inverse function x = g(u) is obtained merely by rotating the original graph through an angle of 180° about the dotted line (Fig. 154), so that the positions of the x-axis and the u-axis are interchanged. The new position of the graph will depict x as a function of u. In its original position the graph shows u as the height above the horizontal x-axis, while after the rotation the same graph shows x as the height above the horizontal u-axis.

image

Fig. 154. Inverse functions.

The considerations of the preceding paragraph may be illustrated for the case of the function

u = tan x.

This function is monotone for − π/2 < x < π/2 (Fig. 152). The values of u, which increase steadily with x, range from – ∞ to + ∞; hence the inverse function,

x = g(u),

is defined for all values of u. This function is denoted by tan-1 u or arc tan u. Thus arc tan(1) = π/4, since tan π/4 = 1. Its graph is shown in Figure 155.

image

Fig. 155. x; = arc tan u

4. Compound Functions

A second important method for creating new functions from two or more given ones is the compounding of functions. For example, the function

image

is “compounded” from the two simpler functions

image

and can be written as

u = f(x) = h(g[x]) (read, “h of g of x”).

Likewise,

image

is compounded from the three functions

image

so that

u = f(x) = k(h[g(x)]).

The function

image

is compounded from the two functions

image

The function f(x) is not defined for x = 0, since for x = 0 the expression 1/x has no meaning. The graph of this remarkable function is obtained from that of the sine. We know that sin z = 0 for z = kπ, where k is any positive or negative integer. Furthermore,

image

if k is any integer. Hence

image

If we set successively

k = 1, 2, 3, 4, · · ·,

then, since the denominators of these fractions increase without limit, the values of x for which the function sin (1/x) has the values 1, –1, 0, will cluster nearer and nearer to the point x = 0. Between any such point and the origin there will still be an infinite number of oscillations of the function. The graph of the function is shown in Figure 156.

image

Fig. 156. Image.

5. Continuity

The graphs of the functions so far considered give an intuitive idea of the property of continuity. We shall give a precise analysis of this concept in §4, after the limit concept has been put on a rigorous basis. But roughly speaking, we say that a function is continuous if its graph is an uninterrupted curve (see p. 310). A given function u = f(x) may be tested for continuity by letting the independent variable x move continuously from the right side and from the left side towards any specified value x1. Unless the function u = f(x) is constant in the neighborhood of x1, its value will also change. If the value f(x) approaches as a limit the value f(x1) of the function at the specified point x = x1, no matter whether we approach X1 from one side or the other, then the function is said to be continuous at X1. If this holds for every point x1 of a certain interval, then the function is said to be continuous in the interval.

Although every function represented by an unbroken graph is continuous, it is quite easy to define functions that are not everywhere continuous. For example, the function of Figure 157, defined for all values of x by setting is discontinuous at the point x1 = 0, where it has the value – 1. If we try to draw a graph of this function, we shall have to lift our pencil from the paper at this point. If we approach the value x1 = 0 from the right side, then f(x) approaches +1. But this value differs from the actual value, – 1, at this point. The fact that – 1 is approached by f(x) as x tends to zero from the left side does not suffice to establish continuity.

f(x) = 1 + x  for x > 0

f(x) = –1 + x for x ≤ 0

image

Fig. 157. Jump discontinuity.

The function f(x) defined for all x by setting

f(x) = 0 for x ≠ 0, f(0) = 1,

presents a discontinuity of a different sort at the point x1 = 0. Here both right- and left-hand limits exist and are equal as x approaches 0, but this common limiting value differs from f(0).

Another type of discontinuity is shown by the function of Figure 158, at the point x = 0. If x is allowed to approach zero from either side, u tends to infinity; the graph of the function is broken at this point, and small changes of x in the neighborhood of x = 0 may produce very large changes in u. Strictly speaking, the value of the function is not defined for x = 0, since we do not admit infinity as a number and therefore we cannot say that f(x) is infinite when x = 0. Hence we say only that f(x) “tends to infinity” as x approaches zero.

image

image

Fig. 158. Infinite discontinuity.

A still different type of discontinuity appears in the function u = sin (1/x) at the point x = 0, as is apparent from the graph of that function (Fig. 156).

The preceding examples exhibit several ways in which a function can fail to be continuous at a point x = x1:

1) It may be possible to make the function continuous at x = x1 by properly defining or redefining its value when x = x1. For example, the function u = x/x is constantly equal to 1 when x ≠ 0; it is not defined for x = 0, since 0/0 is a meaningless symbol. But if we agree in this case that the value u = 1 shall also correspond to the value x = 0, then the function so extended becomes continuous for every value of x without exception. The same effect is produced if we redefine f(0) = 0 for the function defined at the bottom of the preceding page. A discontinuity of this kind is said to be removable.

2) Different limits may be approached by the function as x approaches x1 from the right and from the left, as in Figure 157.

3) Even one-sided limits may not exist, as in Figure 156.

4) The function may tend to infinity as x approaches x1, as in Figure 158.

Discontinuities of the last three types are said to be essential; they cannot be removed by properly defining or redefining the function at the point x = x1 alone.

Exercises: 1) Plot the functions image and find their discontinuities.

2) Plot the functions image and image and verify that they are continuous at x = 0, if one defines u = 0 for x = 0, in both cases.

*3) Show that the function arc image has a discontinuity of the second type (jump) at x = 0.

*6. Functions of Several Variables

We return to our systematic discussion of the function concept. If the independent variable P is a point in the plane with coördinates x, y, and if to each such point P corresponds a single number u—for example, u might be the distance of the point P from the origin—then we usually write

u = f(x, y).

This notation is also used if, as often happens, two quantities x and y appear from the outset as independent variables. For example, the pressure u of a gas is a function of the volume x and the temperature y, and the area u of a triangle is a function u = f(x, y, z) of the lengths x, y, and z of its three sides.

In the same way that a graph gives a geometrical representation of a function of one variable, a geometrical representation of a function u = f(x, y) of two variables is afforded by a surface in the three-dimensional space with x, y, u as coördinates. To each point x, y in the x, y-plane we assign the point in space whose coördinates are x, y, and u = f(x, y). Thus the function image is represented by a spherical surface with the equation u2 + x2 + y2 = 1, the linear function u = ax + by + c by a plane, the function u = xy by a hyperbolic paraboloid, etc.

A different representation of the function u = f(x, y) may be given in the x, y-plane alone by means of contour lines. Instead of considering the three-dimensional “landscape” u = f(x, y), we draw, as on a contour map, the level curves of the function, indicating the projections on the x, y-plane of all points with equal vertical elevation u. These level curves are simply the curves f(x, y) = c, where c remains constant for each curve. Thus the function u = x + y is characterized by Figure 163. The level curves of a spherical surface are a set of concentric circles. The function u = x2+ y2 representing a paraboloid of revolution is likewise characterized by circles (Fig. 165). By numbers attached to the different curves one may indicate the height u = c.

image

Fig. 159. Half sphere.

image

Fig. 160. Hyperbolic paraboloid.

image

Fig. 161. A surface u = f(x, y).

image

Fig. 162. The corresponding level curves.

image

Fig. 163. Level curves of u = x + y.

Functions of several variables occur in physics when the motion of a continuous substance is to be described. For example, suppose a string is stretched between two points on the x-axis and then deformed so that the particle with the position x is moved a certain distance perpendicularly to the axis. If the string is then released, it will vibrate in such a way that the particle with the original coördinate x will have at the time t a distance u = f(x, t) from the x-axis. The motion is completely described as soon as the function u = f(x, t) is known.

image

Fig. 164. Paraboloid of revolution.

image

Fig. 165. The corresponding level curves.

The definition of continuity given for functions of a single variable carries over directly to functions of several variables. A function u = f(x, y) is said to be continuous at the point x = x1, y = y1 if f(x, y) always approaches the value f(x1, y1) when the point x, y approaches the point x1, y1from any direction or in any way whatever.

There is, however, one important difference between functions of one and of several variables. In the latter case the concept of an inverse function becomes meaningless, since we cannot solve an equation u = f(x, y), e.g. u = x + y, in such a way that each of the independent quantities xand y can be expressed in terms of the one quantity u. But this difference in the aspect of functions of one and of several variables disappears if we emphasize the idea of a function as defining a mapping or transformation.

*7. Functions and Transformations

A correspondence between the points of one line l, characterized by a coördinate x along the line, and the points of another line l′, characterized by a coördinate x′, is simply a function x′ = f(x). In case the correspondence is biunique we also have an inverse function x = g(x′). The simplest example is a transformation by projection, which—we state here without proof—is characterized in general by a function of the form x′ = f(x) = (ax + b)/(cx + d), where a, b, c, d, are constants. In this case, the inverse function is x = g(x′) = (–dx′ + b)/(cx′ – a).

Mappings in two dimensions from a plane π with coördinates x, y onto a plane π′ with coördinates x′, y′ cannot be represented by a single function x′ = f(x), but require two functions of two variables:

x′ = f(x, y),

y′ = g(x, y).

For example, a projective transformation is given by a function system,

image

where a, b, · · ·, k are constants, and where x, y and x′, y′ are coördinates in the two planes respectively. From this point of view the idea of an inverse transformation makes sense. We simply have to solve this system of equations for x and y in terms of x′ and y′. Geometrically, this amounts to finding the inverse mapping of π′ onto π. This will be uniquely defined, provided the correspondence between the points of the two planes is biunique.

The transformations of the plane studied in topology are given, not by simple algebraic equations, but by any system of functions,

x′ = f(x, y),

y′ = g(x, y),

that define a biunique and bicontinuous transformation.

Exercises: *1) Show that the transformation of inversion (Chapter III, p. 141) in the unit circle is given analytically by the equations x′ = x/(x2 + y2), y′ = y/(x2 + y2). Find the inverse transformation. Prove analytically that inversion transforms the totality of lines and circles into lines and circles.

2) Prove that by a transformation x′ = (ax + b)/(cx + d) four points of the x-axis are transformed into four points of the x′-axis with the same cross-ratio. (See p. 175.)