Calculus: Analytic Geometry and Calculus, with Vectors

144 Functions, limits, derivatives

relations 0 oo = 1, a = oo, and a = - oo are as absurd in the "algebra" of S*
as the symbol a is in the algebra of S. We can be momentarily delighted by the
"algebraic law" which says that oo + x = oo whenever x is a real number, but
general usefulness of the unorthodox "algebra" is greatly impaired by the fact
that the relation y + x = y does not imply that x = 0 because y might be 00
and x might be 416. For present purposes, we do not need substantial informa-
tion about these matters, but a little basic information can be very helpful.
There are circumstances in which - oo and oo are considered to be numbers, but
there are no circumstances in which - oo and oo are real numbers to which we
can apply the algebraic rules (or laws or axioms or postulates) that apply to
real numbers. Whether or not we consider - oo and oo to be numbers, it is
worthwhile to recognize that some of the most convenient terminologies and
notations of modern mathematics are relics of times when the "doctrine of limits"
was based upon visions of a number x galloping toward infinity and becoming
so infinitely great (but still not co) that its reciprocal becomes infinitesimally
small (but still not 0). These infinitesimals of mathematics, like the aether and
phlogiston of physics and chemistry, can now be regarded as mystic absurdities,
but they were hardy concepts having tremendous impacts upon present as well
as past science and philosophy. We can conclude these remarks with another
bit of history. In the good old days when mathematical terminology was incredi,
bly erratic, sane physicists got the habit of saying that a number is "finite" when
they wished to emphasize their idea that it is neither zero nor infinite nor infi-
nitely small nor infinitely large. It will be interesting to see how long physicists
continue to make modern mathematicians shudder by using the word "finite"
to mean "good honest nonzero noninfinite number, with no nonsense." The
physicists have good intentions, but mathematicians consider zero to be a finite
number, with no nonsense.

3.4 Continuity This section contains information about functions

and limits that we will need. Our first task is to obtain a full under-

standing of the following definition.

Definition 3.41 .4 function f is continuous at xo (or at the point with

coordinate xo, or at the point xo) if

lim f(x) = f(xo).

X-- ze

The assertion that f is continuous at xo is nothing more nor less than the

assertion that f(x) is near f(xo) whenever x is near xo. It means that to

each e > 0 there corresponds a b > 0 such that

(3.42) 1 AX) - f(xo)I < e Ox - XOI < S).

The definition implies that f cannot be continuous at xo unless f(xo)

exists, that is, unless xo belongs to the domain of f. In case f(xo) exists,

the first inequality in (3.42) automatically holds whenx = xo and we do

not need to bother with the restriction x 5 xo that appears in the defini-