244 Chapter 5 • Continuous Functions
- The characteristic function of a set A of real numbers defined by
XA ( x) = {^1 ~f x E A ; }. For each of the following intervals I , determine
0 if x tJ. A.
the points where xr(x) is not continuous from the left, and the points at
which xr(x) is not continuous from the right:
(a)I=(a,b)
(c)I=[a,b)
(e) I= (-oo,a)
(g) I= (a, +oo)- Give an example of
(b) I= [a, b]
(d)I=(a,b]
(f) I= (-oo,a]
(h) I= [a, +oo)(a) a function that has a removable discontinuity at O;
(b) a function that has removable discontinuities at 0, 1, and 2;
(c) a function that has a removable discontinuity at every natural num-
ber.- Prove that if f has a r emovable discontinuity at an interior point xo of
its domain, then f is continuous neither from the left nor from the right
at xo - Prove Theorem 5.2.17 (b).
- Prove Theorem 5.2.17 (c).
10. Prove Theorem 5.2.17 (d).- Prove Theorem 5.2.18.
- Prove that Thomae's function defined in 5.1.12 has only removable discon-
tinuities (at every rational number) , and that the zero function "removes"
them. (See Exercise 5.1.30.) - Prove Theorem 5.2.20 for the monotone decreasing case.
- Prove that if a function f : V(f) ---t JR is monotone on A ('.;; V(f) and
lim f(x) exists at an interior point x 0 of A , then f is continuous at x 0.
x~xo - Prove that if f : (a, b) ---t JR is continuous, monotone, and bounded on
(a, b), where a < b, then f can b e defined at a and b in such a way that
the extended f: [a , b] ---t JR is continuous and monotone on [a, b].
16. Prove that if a function f : I ---t JR is monotone on an interval I , then f
can b e redefined at the points of I where it is discontinuous in such a way
that the redefined function is continuous from the left everywhere on I.
[The same is true if "left" is replaced by "right." ]