## Description

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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