It may be that limit f (c 6)= f (c) only for -1-6, then f(x) is named continuous right of c; or that Lim. f(c -1- cf)— f(c) only for then f(x) is named continuous left of c. Only when f(x) is continuous both nght and left of c 1f(c) being the same], is f(x) continuous at c.
If f(x) be continuous at all points (values of x) right of a and left of b, it is named con tinuous in [a, The infinitesimal [c — a, c a] is called the (immediate) vicinity (or neighborhood) of c.
A change in the value of a v is conveniently denoted by dv, read difference-v or Delta-v; hence dx and dy will denote corresponding (simultaneous) differences or changes in x and y.
If now
f(x) be continuous in [a, b], we may cut this latter up into finite sub-intervals, x, each so small that the oscillation of y in each shall be
Continuity is the supreme functional property with which the Calculus is concerned. Sine and cosine are everywhere continuous, but tan x is discontinuous for x where tan x drops from + co to — oo . Similarly x-1 a at x— a, an extremely important discontinuity.
So 1 er--1) + 1) is discontinuous at x = 0, leaping from —1 to 1; the discontinuity is 2. It is generally assumed that Continuity holds throughout the Processes of Nature.
Again, y 1 sin —is not defined for x = 0, but whatever value be assigned it there, it remains discontinuous, since sin 1 — vibrates infinitely fast between + 1 and —1 for 0.—Again, f(ci-a) may approach a limit for a vanishing, yet not approach f (c). Thus, let f + 1 — . . . , a decreasing geometrical series, ratio (1 + hence Lim. f(x) =1 + x'. Then as x=0, f(x)-1, Lim. f 1; but for x=0, f(x)=f(0)=0.—There are many immediate consequences of continuity, which we have no space to discuss here, such as: A function continuous in [a, b] attains its upper and lower limits (its maximum and minimum) it also assumes at least once every value be tween f (a) and f(b),— a property, however, not peculiar to continuous functions (Darboux).
The notion oŁ function is at once extended to several variables, u=f(x, y, . .), one- or many valued, algebraic or transcendental, etc., as be fore. Here each variable, as x, has its range or in terval [a, a'] ; so y its [b, b'], etc. All possible sets of values (x, y ...) form an assemblage or
the Domain (D) of variation. Any set (or point) for which any variable has an extreme or border value, as a or a, b or b', is a border point; the assemblage of all such is the border or contour of D. A simple geometric depiction of D in rectangular co-ordinates for only two variables, x and y, would be a rectangle with sides x=a, x=a, y=b' ; of three variables, x y, z, it would be a cuboid bounded by the planes x=a, x=a', y=b, y=b', z=c, etc. The point (x, y) or (x, y, 2) may be anywhere in or on the rectangle or cuboid. Such a D may be thought cut up into elements, infinitesi mal rectangles or cuboids. Suppose any point within an element. If now f(x, y, ...) approaches f ) as limit, as point (x, y, ...) approaches point (as, b1, ...), no matter how, then f(x, y,...) is called con tinuous at (a,, bi, ...). This amounts to saying that the oscillation of f shrinks toward zero as the element contracts, no matter how, about the point ; that is, infinitesimal function changes correspond to any and all infinitesimal argument-changes in the immediate vicinity of the point.
Any f(x, y, . . .) is called continuous within D when continuous at every point in D, border included; but on this border, as x, y, . approach a, b,, . . the point must not get without D. An f is continuous in the (imme diate) vicinity of a point, when continuous within an infinitesimal D including that point.
In general, theorems holding for functions of one variable may be extended. with proper mod ifications, to functions of several variables.
In the study of functional dependence, the main subject of scientific inquiry, it is of first importance to know how corresponding changes in the magnitudes are related. To discover this, we form the quo tient of corresponding differences, — called Jr' Difference-Quotient (DQ). In general, it is very complex, but breaks up into two parts, one independent of dx, the other vanishing with dz. The first is the important part and is named Derivative (D) or Differential Coeffi cient (DC). More formally, if y=f (x) be a unique continuous function of x in [a, b], and x be any point therein, and if dy dx dx approaches a limit as dx approaches 0 no matter how, then that limit is called Derivative (D) of f (x), as to x, at the point x. If f(x) has a D at every point of fa, b1, the assemblage of theni forms a new function, the Derivative of f(x) for [a, b],' which we may write f' (x) with Lagrange, or Df(x) with Cauchy. Hence df(x) =--f'(x) +6.