16 1. The Regulated and Riemann Integrals

has a derivative except at a finite set. Suppose that f, g ∈ S

and g([a, b]) ⊂ [a, b]. Prove that h(x) = f(g(x)) is in S.

(3) Let f : [a, b] → C be a complex-valued function and sup-

pose its real and imaginary parts, u(x) = (f(x)) and

v(x) = (f(x)), are both continuous. We can then define

the derivative (if it exists) by df/dx = du/dx + idv/x and

the integral by

b

a

f(x)dx =

b

a

u(x) dx + i

b

a

v(x) dx.

(a) Prove that if F : [a, b] → C has a continuous derivative

f(x) then

b

a

f(x) dx = F (b) − F (a),

i.e., the fundamental theorem of calculus holds.

(b) Prove that, if c ∈ C and F (x) =

ecx

for x ∈ [a, b], then

dF/dx = cecx. Hint: Use Euler’s formula:

eiθ

= cos θ + i sin θ

for all θ ∈ R.

(c) Prove that, if c ∈ C is not 0, then

b

a

ecx

dx =

ecb

−

eca

c

.

1.7. The Riemann Integral

We can obtain a larger class of functions for which a good integral

can be defined by using a different method of comparison with step

functions.

Suppose that f(x) is a bounded function on the interval I = [a, b]

and that it is an element of a vector space of functions which contains

the step functions and for which there is an integral defined satisfying

properties I–V of §1.2. If u(x) is a step function satisfying f(x) ≤

u(x) for all x ∈ I, then monotonicity implies that if we can define

b

a

f(x) dx it must satisfy

b

a

f(x) dx ≤

b

a

u(x) dx.

This is true for every step function u satisfying f(x) ≤ u(x) for all

x ∈ I. Let U(f) denote the set of all step functions with this property.