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
u(x) dx + i
(a) Prove that if F : [a, b] → C has a continuous derivative
f(x) dx = F (b) − F (a),
i.e., the fundamental theorem of calculus holds.
(b) Prove that, if c ∈ C and F (x) =
for x ∈ [a, b], then
dF/dx = cecx. Hint: Use Euler’s formula:
= cos θ + i sin θ
for all θ ∈ R.
(c) Prove that, if c ∈ C is not 0, then
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
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
f(x) dx it must satisfy
f(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.