Show that f notin r 0 1 find a nonconstant monotonically

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, b]. Show that there is ξ ∈ [a, b] such that

_a∫^b f(x)g(x)dx = f(ξ) _a∫^bg(x) dx.

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 = {x₀, ··· , x_n} be an appropriate partition of [a, b] and define

h(x) = (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 ∩ [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

             x              if x ∈ Q ∩ [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 ([0, 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 f ob [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 ≡ 0 on [a, b].