Calculus: Analytic Geometry and Calculus, with Vectors

146 Functions, limits, derivatives

function, like fl, is continuous except when x = a and x = b. However,

f2 has left-hand continuity at a and right-hand continuityt at b.
Definition 3.45 4 function f is continuous over an interval a 5 x 5 b
if it is continuous at each xo for which a < xo < b and, in addition, has
right-hand continuity at a and left-hand continuity at b.
The definitions of this section are designed to be useful in discussions of
examples of functions, and we begin by looking at examples of functions.
Let g be the function, defined for x 0 0, for which

(3.451) g(x) =x (x ; 0).

This function is continuous at each xo 0 0 because, when xo 0, our

theorems on limits imply that

(3.452) mx0 g(x) = lx'm 1x=

1

lim x

=^1

xo =g(xo)

x-.xo

However, g cannot be continuous at 0, because g(O) is undefined and there

is no possibility of having lim g(x) = g(0). We say that g is discon-

x- O

tinuous at 0. Now let h be the function defined over - m < x < oo

(this means merely that the domain of h is the entire set of numbers) by

h(0) = 0 and

(3.453) h(x) = z (x s 0).

This function, like g, is continuous at each xo F6 0, but this time h(0)

exists and there is no possibility of having lim h(x) = h(0), because

lim h(x) does not exist. Let w (omega) be the peculiar function for

x-.o

which w(0) = 1 and w(x) = 0 when x 5-4- 0. For this function both

w(0) and lim co(x) exist, but the function is discontinuous at 0 because

(3.454) lim w(x) = 0 0 1 = CO(O).

x-.o

t It is to be expected that some readers, particularly those more interested in applied

mathematics than in pure mathematics, may feel that matters now being considered are

much too theoretical to have practical interest. Some people know, and others can learn,

that when a battery has its terminals connected to appropriate electrical hardware, it almost

instantly produces an electromotive force (the kind of a force that pushes or pulls electrons

around) which we may, for present purposes, suppose to have the constant value 1. When

the battery is not connected, the electromotive force produced by it is 0. Thus, batteries

which are connected over some time intervals, and disconnected over other time intervals,

produce electromotive forces that are, as functions of time, very closely approximated

by step functions such as those we have been considering. The discontinuous functions

are introduced to simplify problems, not to complicate them. This is one of the reasons

why persons interested in applications of science must recognize existence of discontinuous

functions.