(a)
Construct
an
example
of
a
function
f:[0,1]
→
which
is
continuous nowhere and a function g:[0,1]
→
which is continuous
on all of [0,1], such that m({x: f(x)
≠
g(x)}) = 0.
(b)
Let <q
n
> be an enumeration of all the rational numbers in the
interval [0,1].
Define f:[0,1]
→
by the following formula:
f(x) =
2
n
.
q
n
≤
x
Here, of course, the sum is over the indices n such that q
n
is no
larger than x.
Note that since the geometric series
2
n
dominates
the sum defining f, convergence is not a problem.
Show f is continuous at each irrational number in [0,1] and
discontinuous at each rational number in [0,1].
Can you find a continuous function g:[0,1]
→
such that the
measure of the set {x: f(x)
≠
g(x)} is zero?
Proof??
Hint:
(1) For (a) there are a couple of very easy and obvious
functions that do the job.
(2) For (b), it helps to observe that the discontinuities
are jumps since f is increasing.
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View Full DocumentMAA5616/FINAL EXAM/PART C
Autumn, 1997
Page 6 of 6
6.
(a) Let f
n
(x) = n
1
χ
[n,n]
(x) for n
ε
and x
ε
.
Show {f
n
}
converges uniformly to f(x)=0o
n .
(b)
With
proof
determine
whether
there
is
a
Lebesgue
integrable function g defined on
such the g(x)
≥
f
n
(x) for every
x
ε
and n
ε
.
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 Spring '13
 Ritterd
 Lebesgue integration, Lebesgue, MAA5616/FINAL EXAM/PART

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