
“real: chapter_05” — 2011/5/22 — 22:55 — page 45 — #45
Limits and Continuity 5-45
We now observe that
d
2
(F(x), F(y)) ≤ d
2
(F(x), f (x
n
))+d
2
(f (x
n
), f (y
n
))+d
2
(f (y
n
), F(y)).
(5.5)
Using (5.3) to (5.5) and choosing n large, we see that
d
2
(F(x), F(y))< if d
1
(x, y)<
δ
3
.
This proves the required uniform continuity of F.
Further, if f is an isometry, then clearly f is uniformly continuous
and further (5.2) gives
d
2
(F(x), F(y)) = lim
n→∞
d
2
(f (x
n
), f (y
n
))
= lim
n→∞
d
1
(x
n
, y
n
)
= d
1
(x, y)
showing that F is also an isometry.
SOLVED EXERCISES
1. Study the continuity of the following functions:
(a) f (x) =
0ifx is irrational or x = 0,
1
q
if x =
p
q
, p ∈ Z, q ∈ N and p, q are ...