Bridge to Abstract Mathematics: Mathematical Proof and Structures

130 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4

Many of the most familiar functions lead always to the Type I limit

scenario and thus cause little difficulty in the context of limit problems. In

particular, polynomial functions are continuous everywhere and rational

functions are continuous everywhere except at those values of a that make

their denominator zero. Also, familiar transcendental functions such as

ex, In x, sin x, and cos x are continuous wherever defined. Other, less

conventional, examples are:

which is continuous everywhere

x- 1, except at x = 0.

which is continuous everywhere, even at

1, x rational

h(x) = - 1, x irrational which is continuous nowhere.

x, x rational which is continuous at one point
k(x) =
0, x irrational ' only namely, x = 0.
Continuity of a function f at a point a requires three things: (1) a must be
in the domain off, often stated f is dejned at a or f (a) exists. (2) lirn,,, f (x)
must exist. (3) The number lim,,, f (x) must be the same as the number
f(a). We will discuss the reasons for the noncontinuity of functions such
as the preceding f, h, and k, at some or all points of their domain, after our
introduction to categories I1 and 111.

Type IT. Under this category, we consider the possibility that lim,,, f(x)

might exist, but have a value other than f(a). Within this category f(a)

itself may or may not be defined. This is probably the category on which

most students have the most tenuous hold, after their first exposure to

limits, so we will look at it carefully.

Graphically, this situation is characterized by an otherwise continuous

curve with a point missing at x = a. In this situation f is said to have a

removable discontinuity at a. The name is apt; we could "remove" the dis-

continuity simply by redefining f at one point, or graphically, by plugging

the hole in the curve. We now give examples to illustrate the two possi-

bilities (1) f (a) exists and (2) f is not defined at a.

(a) A typical example for which lim,,, f(x) and f(a) both exist, but

are not equal, is a function defined by two rules, one rule for x = a and

another for all other values of x; that is, x # a. For example, the function

is not continuous at x = 2, even though lirn,,, f(x) exists (Note: This limit

equals 7. When graphing this function, notice that the limit as x approaches

2 is the y component of the "missing point.") and even though f (2) is defined