Bridge to Abstract Mathematics: Mathematical Proof and Structures

132 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4

THE EPSILON-DELTA DEFINITION AND

ITS RELATIONSHIP TO TYPES I, 11, AND I11

By definition

L = lim f(x) if and only if, for every E > 0, there exists 6 > 0 such

x-ra

that whenever 0 < Ix - a1 < 6, then 1 f(x) - LI < E

In this definition we assume that f is defined in an open interval containing
a. The logical complexity of this definition is considerable. In particular,
for a fixed value of a, this expression is a compound propositional function
in three variables, E, 6, and x, involving the connective +, the most difficult
connective from the point of view of intuition. Furthermore, the quantifi-
cation of the three variables ig mixed quantification, again, the most difficult
case intuitively. Not surprisingly, this rigorous definition was formulated
(by the German mathematician Karl Weierstrass) a full two hundred years
after the intuitive idea of limit had been used by both Newton and Leibniz
in their independent invention of the derivative concept, and thereby the
calculus itself, around 1675. Even after understanding the geometric signif-
icance of the inequalities 0 < lx - a1 < 6 and 1 f(x) - LI < E [the former
determining the "interval minus one point" (a - 6, a + 6) - (a}, along the
x axis, the latter the interval (L - E, L + E) along the y axis], we still have
difficulty in seeing the connection bet ween the epsilon-del ta definition and
the results obtained in actual calculations of specific limits. See Figure 4.1,

Figure 4.1 Epsilon and delta bands.

y = L The vertical 6 band

a-6toa+6

X

from