Answers & Hints for Selected Exercises 659
Suppose I is continuous at x 0. Then 38 > 0 3 '<Ix E V(f), Ix - xol <
8 =::;. ll(x) -l(xo)I S 1 =::;.I ll(x)l - ll(xo)l I S 1 (Thm. 1.2.15 (c)) =::;. ll(x)I S
ll(xo)I + 1.
( =::;.) Suppose I is continuous on JR, let Ube open, and x 0 E 1-^1 (U). Then
l(xo) EU, which is open, so 3e > 0 3 l(xo) E N"(f(xo)) i;;:; U. By continuity,
38 > 0 3 x E No(xo) =::;. ll(x) - l(xo)I < e =::;. l(x) E N"(f(xo)) i;;:; U =::;. x E
1-^1 (U). Thus, '<lxo E 1-^1 (U), 38 > 0 3 N 0 (xo) i;;:; 1-^1 (U) .. ·. 1-^1 (U) is open.
( {::::) Suppose that 'ti open set U, 1-^1 (U) is open. Let x 0 E JR and e >
Then 1-^1 (N"(f(xo)) is open. Since xo E 1-^1 (N"(f(xo)), which is open,
38 > 0 3 No(xo) i;;:; 1-^1 (N"(f(xo)); i.e., I (N 0 (xo)) i;;:; N"(f(xo)). This means
Ix -xol < 8 =::;. ll(x) - l(xo)I < e. Therefore, I is continuous at xa.
Suppose l ,g: A-+ JR are continuous and l(x) = g(x) for all x in a dense
subset B of A. Define h(x) = l(x) - g(x). Then h is continuous on A and
'<Ix EB, h(x) = 0. By Ex. 28, '<Ix EA, h(x) = 0.
I= the Dirichlet function, A= Q, B = Qc.
EXERCISE SET 5.2- ( =::;.) Suppose I is continuous from the left at Xo. Let { Xn} be a sequence
in V(f) n (-oo, xo) converging to xa. Let e > 0. Then 38 > 0 3 '<Ix E V(f),
xo - 8 < x < xo =::;. ll(x) - l(xo)I < e and 3no EN 3 n ~no=::;. xo - 8 < Xn <
Xo =::;. ll(xn) - l(xo)I < €. .". l(xn)-+ l(xo).
( {::::) Suppose I is not continuous from the left at x 0. Modify the procedure
used in the proof of Thm. 4.1.9 to find a sequence {xn} in V(f) n (-oo,x 0 )
converging to Xo such that l(xn) ft l(xo). - (a) Let xo tt Z. Then 38 > 0 3 lxJ is constant on N 0 (xo), say lxJ = c. Then,
X-+Xo lim lxJ=c=lxoJ.
(b), (c) Let xo E Z. Then 38 > 0 3 xo - 8 < x < xo =::;. lxJ = xo - 1 and
xo < x < Xo + 8 =::;. lxJ = xa. Thus,
lim lxJ = xo - 1 =f lxoJ while lim lxJ = xo = lxoJ.
x--+x 0 x--+xt
- not cont. from left at not cont. from right at
(a) b a
(b) a b
(c) a,b nowhere
(d) nowhere a,b
(e) a nowhere
(f) nowhere a
(g) nowhere a
(h) a nowhere