Find a nonconstant monotonically increasing function g on 0

Question

1. (a) If f is continuous on [a, b] and _a∫^b|f (x)| dx = 0, then show that f ≡ 0 on [a, b].

(b) Suppose g a non-constant monotonically increasing function on an interval [a, b] and let f be continuous on (a, b] such that

_a∫^b f dg = 0.

Show that f (x₀) = 0 for some x₀ ∈ [a, b]. Give an example to show that f need not be identically zero on [a, b].

2. Let a < c < b, and g be monotonically increasing on [a, b]. Suppose that f ∈ R([a, c), g) and f ∈ R.([c, b], g).

(i) Use Riemann's Integrability Criterion to show that f belongs to R([a, b], g).

(ii) Show that _a∫^b f dg = _a∫^c f dg + _c∫^b f dg

3. Suppose f:[a, b] → R is a continuous function and g: [a, b] → R is a non-negative Riemann integrable function on [a, bJ. Show that there is ξ ∈ [a, bJ such that

_a∫^b f (x)g(x)dx = f (ξ) _a∫^b g(x) clx

Hint: First explain why fg ∈ R([a, b]). Then use the Intermediate Value Property of f.

4. Show that f ∈ R([0,1]), where f is defined by

          1/n             if x = -m/n for some m, n ∈ N with gcd(m, n) = 1
 f(x) =

           0               otherwise.

5. Let f ∈ R([a, b], g). Given ∈ > 0, show that there is a continuous function h on [a, b] such that

_a∫^b |f(x) - h(x)|² dg(x) < ∈

Suggestion. Let P = {x_o, .........., x_n} be an appropriate partition of [a, b], and define

h(t) = (x_j - t)/Δx_j.f(x_{j -1}) + t - x_j-1/Δx_j . f(x_j) ( x_j-1 ≤ t ≤ x_j.

6. Let h be a positive continuous function on [0,1]. Define f on [0,1] by

          h(x)       if x ∈ Q n [0,1]
f(x) =
            0         if x ∈ [0, 1] \ Q.
Show that f ∉ R,([0, 1]).

Hint: Assume the contrary. Express f in terms of the Dirichlet function on [0, 1], which is known to be not Riemann integrable.

7. Consider the function defined on [0, 1] by

                  if x ∈ Q n [0,1]
f (x) =
                  0 if x ∈ [0,1] \ Q.

(a) Show that f ∉ R.([0 , 1]). 

(b) Find a nonconstant monotonically increasing function g on (0, 1] such that f ∈ R([O, 1],g), and provide a proof for your assertion.

8. (a) Let g be a monotonically increasing function on [a, b] such that

_a∫^b f.dg = 0

for every nonconstant continuous function g on [a, b]. Show that g is a constant on [a, b.]

(b) Let f : [a, b] → R be a continuous function such that

_a∫^b f.dg = 0

for every monotonically increasing function g on [a, b]. Show that f
f ≡ 0 on [a, b].