MAA5616/FINAL EXAM/PART C
Autumn, 1997
Page 2 of 6
2.
Define f:[0,1]
→
by f(x) = x if x is irrational or x = 0,
and f(x) = x + (1/n) if x
≠
0 is rational and x = m/n in lowest
terms.
(a) Give an
ε

δ
proof that f is not continuous at each rational
number x
ε
(0,1].
(b) Give an
ε

δ
proof that f is continuous at each irrational
number x
ε
(0,1].
(c) Compute the value of the Riemann integral
1
R
∫
f(x) dx.
0
Hints:
(1) For each positive integer n there are only finitely
many rational numbers with denominators less than n in each finite
open interval.
(2) At some point Proposition 4.7 might be useful.
(3) If you want, you may use the Fundamental Theorem of
Calculus to evaluate a certain Riemann integral you encounter along
the way.
Be careful to ensure you have satisfied all the
hypotheses, however.
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MAA5616/FINAL EXAM/PART C
Autumn, 1997
Page 3 of 6
3.
Definition: A realvalued function f is
upper semicontinuous
at x
if, and only if, for each
ε
> 0, there’s a
δ
> 0, such that
for each t in the domain of f, if t  x <
δ
, then
f(t) < f(x) +
ε
.
Prove Dini’s Theorem: Suppose <f
n
> is a sequence of upper
semicontinuous functions defined on a closed and bounded subset A
of
with f
n
(x)
→
0 as n
→ ∞
for each x
ε
A, and f
n
(x)
≥
f
n+1
(x)
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 Spring '13
 Ritterd
 Lebesgue integration, Lebesgue, MAA5616/FINAL EXAM/PART

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