Real Variable

Functions Defined on an IntervaL—On ac count of the generality of the definition, prac tically no theorems are known which apply to all functions on a given interval. One of the most general theorems known is obtained by introducing the restriction that on a certain interval, f (x) shall have an upper bound. By this it is meant that there shall be a number, B, such that for every x on the given interval f (x)G B. From the principle of continuity it is dear that a function having an upper bound upon an interval, i, has a least upper bound B., and by aid of the Heine-Borel theorem, it is easily proved that there is at least one point, xi, on the interval, such that for every interval of i which includes xi, B. is the least upper bound of f(x). This is a theorem of Weierstrass.' Another broad class of functions is that of the monotonic functions.' A function, f(x), is monotonic increasing if whenever xi

In antithesis to the monotonic functions stand oscillating functions, those such that on the interval under consideration there are three values of the independent variable, xi, s,, xi, aaf CVO, or f(xi) >f (x2), (x.) < f (xi). There exist functions wh.ich oscillate on every interval; indeed, a function may oscillate on every interval and still be continuous and have a continuous derivative' The definition of conti nuity isgiven below and of derivation in the article CALCULUS. In most investigations hitherto the functions considered have had only a finite number of oscillations, i.e., there would exist a finite set of points such that on the intervals between successive points the function would not oscillate. It is to be noted that a non-oscillating function is not necessarily'mono tonic ; there may be intervals upon which it is a constant.

The word oscillation is also used to denote a measure of the total amount of variation of a function on a given interval, that is, the differ ence between the upper and lower bounds of f (s). This is a satisfactory measure in case f(x) is monotonic. In other cases, especially in case of functions with oscillation on every interval, this scheme is naturally extended as follows: Let xi, xi, xi, • • • , xn--1 be any fi nite set of points on the interval ab in the order a< xi< xi

But R is, except for non r=1 oscillating functions, different from f (b)—f (a). For every infinite set of points xi, x2, . . . ,

x22--1, there is a value of R. The least upper bound, V, of the set of all possible numbers, R, is called the variation of f(x) on the interval ab. A function for which. V is finite is called a function of limited variation; such a function can always be expressed as the difference be tween two monotonic functions. Functions of limited variation were first defined and studied by C. Jordan in connection with the theory of lengths of curves.

Theory of Limits.— Turning from the de scription of a function by its properties on an interval to a consideration of its appearance in the immediate neighborhood of a point, we are led to consider the notions of limits and of continuity. If a is a limit point of a set of points [x], then the variable x is said to approach a on the set [x]. In case all the values of x are>a, the approach is said to be from above, and in the opposite case, from low. Let f (x) be a function which is defined and bounded on an interval ab, a a, such that Ix—al < 6 and If(x)—yel< e. A number, ye, satisfying this condition is called a value approached by f(x) as x approaches a from above. If in every in terval Fre of ab, f(x) is without an upper bound, f(x) is said to approach the value of + co as x approaches a from above; if f(x) is without a lower bound in every interval ax, f(x) ap proaches — Thus we have the highly im portant theorem that as x approaches any finite value from above (and by parity of reasoning, from below) t(x) approaches at least one value, finite or infinite.

A finite value approached by f(x) as x ap proaches + m is a number, ye, such that for every pair of positive numbers, e and 6, there is a value of x such that x> d and If (x) — 31.1< e, f(x) approaches + CO as x approaches w if, for every a and 6, there is a value of x such that x>6 and t(x)> e. The definition of approach to — COIs analogous. In the case where x approaches a finite value, x., from above or below or from both sides, f(x) ap proaches y. if for every pair of positive num bers, e and 6, there is a value of x (x+xe), such that Ix —xel < 6 and If(x)—f(x.)1 < e. The theorem of the last paragraph now extends to all cases: if x approaches any value finite or infinite, t(x) approaches at least one value, finite or Infinite.