140 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4particular E anywhere in such a proof. We must begin by letting an arbi-
trary positive e be given. This E, although arbitrary (i.e., not a specifically
named number, like 4 or n/3) is fixed at the start and used throughout the
remainder of the proof. Our job in writing the proof is to determine a pos-
itive number 6, in terms of the given E, such that every x within distance
6 of a, other than a itself, has its corresponding f(x) lying within E of L.
In doing this, we are determining geometrically the comparative width be-
tween a given e band, about the line y = L and a corresponding 6 band
about the line x = a, for which the definition of limit is satisfied. Of partic-
ular importance is the fact that 6 depends on e; typically, 6 might be taken
to equal e, or e/2, or the smaller of 1 and e/3. Dependence of this type, a
consequence of the logical structure of the definition of limit, in which 3 fol-
lows V, was highlighted in Article 3.4.
An interesting application of this reasoning is in arguing that
x rational
lirn,,, k(x) = 0 = k(O), where k(x) =
x irrational, so that k is
continuous at zero. Since k is continuous at no other point, k provides us
with an example in which continuity at a point fails to correspond to our in-
tuitive notion of "no break in the graph in the immediate vicinity of x = a."FURTHER REMARKS ON THE TYPE I1 CASE
To conclude this article, we recall the question raised in considering Type
I1 limit problems. How does the epsilon-delta definition of limit yield the
rule of thumb that the "y component of the missing point" serves aslirn,,, f (x) for a function such as f (x) =
this rule of thumb yields the conclusion that lirn,,, f (x) = limi,, 2x + 5 =
11, even though f(3) = 2. Hence f is not continuous at 3, in spite of the
fact that f (3) is defined and lirn,,, f (x) exists.
At the same time as we raise this question, let us ask another question
about the structure of the epsilon-delta definition. Why does the defini-
...-. tion contain the two inequalities 0 c Ix - a1 < 6 rather than just the single
inequality (x - a1 < b? It turns out that the two questions are related to
each other and to a statelgent about which you have probably heard or read
concerning limits, namely, "the value of lirn,,, f(x) is in no way influenced
by the *slue off La, but is instead completely determined by the values
of f(x), for x in the immediate vicinity of a."
Why are we really dealing here with three pieces of the same puzzle?
Let us begin with the second piece, the epsilon-delta definition itself. Sup-
pose the definition of limit were formulatedL = lirn f(x) 9 (Ve (E 0)(36 > O)(Vx)[(lx - a1 < 6) 4 (1 f(x) - LI < E)].
x-ra