
“real: chapter_05” — 2011/5/22 — 22:55 — page 25 — #25
Limits and Continuity 5-25
(note that if 1 ≤ n ≤ N and if x
n
> x, then x
n
> t. Thus each
c
n
(1 ≤ n ≤ N ) will not appear in this summation). This shows that
f (x+) = f (x) for each x ∈ (a, b) \ E.
These arguments clearly show that for x ∈ (a, b) \ E, f (x+) =
f (x−) = f (x) or that f is continuous at x.
On the other hand, if x = x
m
for some m, then for the same choice
of t as before (i.e. x < t < x + δ
2
), we have
f (t) − f (x) =
x≤x
n
<t
c
n
= c
m
+
x<x
n
<t
c
n
< c
m
+
∞
n=N +1
c
n
< c
m
+
showing that f (x
m
+) = f (x
m
) + c
m
. Since c
m
> 0, f is discontinuous
at x = x
m
. This completes the proof.
5.6 UNIFORM CONTINUITY
Let ...