Assignment 5 of Math 5302
1. Let f be a real-valued bounded function on [−1, 1]. Let
α(x) =
0 if −1 ≤ x < 0;
2 if 0 ≤ x ≤ 1.
Assume f is Riemann-Stieltjes integrable with respect to α on [−1, 1]. Show that
(a) f is continuous at 0 from the left.
(b) R 1
−1
f(x)dα(x) = 2f(0).
2. Let f and α be a real-valued bounded functions on [a, b] and α is increasing. Let L(f, α) and
U(f, α) represent the lower and upper Darboux-Stieltjes integral of f with respect to α on [a, b], respectively.
(a) Show that U(f, α) ≤ U(|f|, α).
(b) Is it true that L(f, α) ≤ L(|f|, α)?
3. Let α be a bounded real-valued increasing function on [a, b]. Assume a < c < b and α is continuous
at c. Let
f(x) =
1 if x = c;
0 if x 6= c.
Show directly that f is Darboux-Stieltjes integrable on [a, b] and R b
a
f(x)dα(x) = 0. (Do not use
Theorem 8.16.)
4. Let f and α be real-valued bounded functions on [a, b] and α is increasing on [a, b]. Assume f
is Darboux-Stieltjes integrable with respect to α on [a, b]. Let [c, d] ⊂ [a, b]. Show that f is DarbouxStieltjes integrable with respect to α on [c, d].
5. Let α be a real-valued bounded function on [a, b] and α is increasing with α(a) < α(b). Let
f(x) =
1 if x is rational;
0 if x is irrational.
Show that if α is continuous on [a, b], then f is not Darboux-Stieltjes integrable with respect to α on
[a, b].
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