5.3 Continuity on Compact Sets and Intervals 245- Prove that Theorem 5.2.17 remains true if {f(x) : a < x < c} and
{f(x) : c < x < b} are replaced by {f(r) : r E Q and a < r < c} and
{! ( r) : r E Q and c < r < b}, respectively. State a similar revision of
Theorem 5.2.18. - Suppose a < b. Prove that if f is monotone on [a, b), and lim f(x)
x->b-
exists, then f is bounded on [a, b). State and prove a similar result for
(a, b] and lim f(x).
x-+a+ - Explain and justify the claim that the function f(x) = { xsin ~ ~f x ~ O}
1 if x = 0
is "oscillating" at 0 and is discontinuous at 0 but does not have an oscil-
lating discontinuity at 0. What kind of discontinuity does it have there? - Prove that Theorem 5.2.17 can be extended to infinite intervals:
(a) If f is monotone increasing and bounded on some [a, +oo), then
lim f(x) exists and equals sup{f(x): x 2'. a}.
x->+ oo
(b) If f is monotone decreasing and bounded on some [a, +oo) then
x-+-lim ex:> f(x) exists and equals inf{j(x): x 2'. a}.State corresponding results for f monotone and bounded on (-oo, a].
5.3 Continuity on Compact Sets and Intervals
We begin this section with a subtle point that may at first hardly seem worth
mentioning, but which can lead to subtle errors of thought if ignored. When
discussing continuity of a function on a set it is often important to understand
the role p layed by the declared domain of the function.
REST R I CTING THE DOMAIN OF A FUN CTION
Suppose f: V(f) -+ R Sometimes we are interested in the behavior of the
function only on a subset A of V(f). For a given A ~ V(f), there is a subtle
difference between the following two statements:^8
Statement #1 f : V(f)-+ JR is continuous on A.
Statement #2 f: A-+ JR is continuous (on A).
- Recall that a function consists of two sets and a "rule" f that associates to each member
of the first set (its domain) a m ember of the second set (its codomain). See Appendix B.2.
We often use the same symbol, f , to d enote the function when we restrict its domain to A.