144 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4
- *(a) Give an epsilondelta argument that &,,+ sin (l/x) # 0.
(b) How might an argument given in (a) have to be altered to prove that
&+o+ (A) sin (llx) # O? lim,,, + (A) sin (llx) # O?
(c) Do you think that the kind of argumemt given in (a) ad (b) could be used to
prove that lim,,,+ (x sin (llx)) # O? Do you think that this one-sided limit ex-
ists? If so, what do you think is its value? (See Exercise 13(c), Article 6.1.)
6. Let f be a function defined on the interval [a, a + r) for some r > 0. We
say that f is right-contimmus, or c~n~inuous jiom the right at a if and only if
limx+, + f (x) = f (a).
(a) Give an analogous definition of left-continuous at a
(b) Give examples of functions f (x), g(x) such that:
(i) f is continuous fiom the right, but not the left, at x = 6.
(ii) g is continuous fkom the left, but not the right, at x = 3.
(c) Suppose that lim,,, f(x) exists, but does not equal f(a) (k, f is Type 11 at a).
Is it possible for f to be continuous from either the left or the right at a?
(d) Using the theorem stated in Exercise 4, state and prove a theorem that relates
right and left continuity off at a to ordinary continuity off at a - A function f is said to be contimow on an open intenml I if and only iff is con-
tinuous at each point of I. In terms of epsilons and deltas, f is continuous on I
if and only if
On the other hand, a function f is said to be un$onnly continuous on I if and
only if(a) Explain, in terms of theorems of the predicate calculus, why uniform con-
tinuity on I is a stronger property than continuity on I. That is, why is it true that,
for any function f, iff is uniformly continuous on I, then f is continuous on I?
We can give many examples of functions that are uniformly continuous on
various intervals, for instance, f(x) = Mx + B is uniformly continuous over any
interval, g(x) = x2 is uniformly continuous on (0,2), while h(x) = l/x is uniformly
continuous on 11, oo). On the other hand, there are functions that are continuous,
but not uniformly continuous, on certain intervals [i.e., the converse of the result
in (a) is false]. For example, f(x) = x2 is not uniformly continuous on (0, p) and
h(x) = l/x is not uniformly continuous on (0, I), even though both functions are
continuous on those intervals.
(b) Suppose we are writing proofs, by using epsilons and deltas, that f(x) = x2
is continuous on the intervals (0-2) and (0, oo). Note that f is uniformly continuous
in the first case, but not in the second. Describe the qualitative difference in the
choice of 6, given a positive E, between these two cases. (Recall Example 3 and
Exercises 6 and 7, Article 3.4.)- Let f be defined on some open interval (a, oo) and let L be a real number. We
say L = I&,, f(x) iff(V& > OX3N > O)(Vx)[(x > N) -, (I f(x) - LI < E)]. Geomet-
rically, this means that any e band about the horizontal line y = L, no matter how
narrow, contains every point on the graph off for values of x to the right of some
number (namely, N). Note that, from the logical structure of the definition, N