Assignment 1 of Math 5302
1. Consider f(x) = 2x + 1 over the interval [1, 3]. Let P be the partition consisting of the points
{1, 1.5, 2, 3}.
(a) Compute L(f, P), U(f, P) and U(f, P) − L(f, P).
(b) What happens to the value of U(f, P) − L(f, P) when we add the point 2.5 to the partition?
(c) Find a partition P
0 of [1, 3] for which U(L, P0
) − L(f, P0
) < 2.
2. Let
f(x) =
x if x is rational on [0, 1];
0 if x is irrational on [0, 1].
(a) Find the upper and lower Darboux integrals for f on the interval [0, 1].
(b) Is f integrable on [0, 1]?
3. Let
f(x) =
1 if x =
1
n
for some n ∈ N;
0 otherwise.
Show that f is integrable on [0, 1] and compute R 1
0
f.
4. Let f be a bounded function on [a, b]. Suppose there exist sequences (Un) and (Ln) of upper and
lower Darboux sums for f such that limn→∞
(Un − Ln) = 0. Show f is integrable and R b
a
f = limn→∞
Un =
limn→∞
Ln.
5. Let f be integrable on [a, b], and suppose g is a function on [a, b] such that g(x) = f(x) except for
finitely many x in [a, b]. Show g is integrable and R b
a
f =
R b
a
g.
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