Let g a b rarr r be a continuous and monotonically

Question

(1) Let a < c < b and α ∈ R \ {0}. Define f : [a, b] → R by

                  0,    if x ≠ c

f(x) =

                  α    if x = c.

Let g : [a, b] → R be a be a monotonically increasing function.

i) Show that f → R([a, b], g) if and only if g is continuous at c.

ii) If g is continuous at c, compute _a∫^b f dg.

(2) Let g : [a, b] → R be a continuous and monotonically increasing function, and suppose f ∈ R([a, b], g). Suppose f is redefined at a finite number of points in [a, b] and h is the resulting function. Show that h ∈ R([a, b] g) and

_a∫^b f dg = _a∫^b h dg.

Hint: Use the conclusions of Problem 1 above applied to the difference f - h.

(3) Let f : [0, 1] → R be defined by

                x²            for x ∈ [0, 1] ∩ Q

f(x) =

                0              for x ∈ [0, 1]\ Q.

i) Show that f is continuous only at x = 0.

iI) If 0 ≤ ξ ≤ ς ≤ 1 show that (ς + ξ/2)² (ς - ξ) ≥ ¼ (ς³ - ξ³).

iii) Use the inequality in (ii) to show that f ∉ R([0,1]).

(4) Suppose f is bounded on [a, b] and continuous on (a, b). If g; [a, b] → R is a monotonically increasing function on [a, b] that is continuous at b, show that f ∈ R([a, b], g).

(5) Suppose f ∈ R([0,2], g) where g is defined by

                1              for x ∈ [0, 1)

g(x)

                x              for x ∈ [1, 2].

Define

F(x) =₀∫^xf dg        for x ∈ [0, 2].

Assume that f is continuous at x = 1. Show that F is differentiable at x = 1 if and only if f(1) = 0.

(6) Compute ₀∫¹ (3x² + 2) dg, where

                1              for x = 0

g(x) =

                x+3         for x ∈ (0, 1]