Question

1 Define f : [0, 1] ? R as follows: f(a) = { 0 if a is irrational, 1/n if a is rational and n is the smallest positive integer such that a = m/n for some integer m. Show that f is Riemann integrable and compute ?? f. 2 Suppose f : [a, b] ? R is a bounded function. Prove that f is Riemann integrable if and only if L(?f, [a, b]) = ?L(f, [a, b]). 3 Suppose f, g: [a, b] ? R are bounded functions. Prove that L(f, [a, b]) + L(g, [a, b]) ? L(f + g, [a, b]) and U(f + g, [a, b]) ? U(f, [a, b]) + U(g, [a, b]).

          1 Define f : [0, 1] ? R as follows:
f(a) = {
0 if a is irrational,
1/n if a is rational and n is the smallest positive integer such that a = m/n for some integer m.
Show that f is Riemann integrable and compute ?? f.
2 Suppose f : [a, b] ? R is a bounded function. Prove that f is Riemann integrable if and only if
L(?f, [a, b]) = ?L(f, [a, b]).
3 Suppose f, g: [a, b] ? R are bounded functions. Prove that
L(f, [a, b]) + L(g, [a, b]) ? L(f + g, [a, b])
and
U(f + g, [a, b]) ? U(f, [a, b]) + U(g, [a, b]).

1 Define f : [0, 1] ? R as follows:
f(a) =
0 if a is irrational,
1/n if a is rational and n is the smallest positive integer such that a = m/n for some integer m.
Show that f is Riemann integrable and compute ?? f.
2 Suppose f : [a, b] ? R is a bounded function. Prove that f is Riemann integrable if and only if
L(?f, [a, b]) = ?L(f, [a, b]).
3 Suppose f, g: [a, b] ? R are bounded functions. Prove that
L(f, [a, b]) + L(g, [a, b]) ? L(f + g, [a, b])
and
U(f + g, [a, b]) ? U(f, [a, b]) + U(g, [a, b]).

Added by Alejandro Y.

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