1.6. The Fundamental Theorem of Calculus 13

(3) Give an example of a sequence of step functions which con-

verges uniformly to f(x) = x on [0, 1].

(4) Prove that the collection of all regulated functions on a

closed interval I is a vector space which contains the con-

stant functions.

(5) Prove that the regulated integral, as given in (1.5.3), satisfies

properties I–V of §1.2.

(6) Suppose an integral satisfying properties I–V of §1.2 has

been defined for all functions f : [a, b] → R in some vector

space of functions V. Prove that if {fn} is a sequence of

functions in V which converges uniformly to f ∈ V, then

lim

n→∞

b

a

fn(x) dx =

b

a

f(x) dx.

(7) Suppose f : [0, 1] → R is continuous on (0, 1). Prove there is

a sequence of step functions {fn} which converge pointwise

to f on [0, 1].

(8) ( ) Prove that f is a regulated function on I = [a, b] if and

only if both of the limits

lim

x→c+

f(x) and lim

x→c−

f(x)

exist for every c ∈ (a, b). (See section VII.6 of Dieudonn´e

[D].)

1.6. The Fundamental Theorem of Calculus

The most important theorem of elementary calculus asserts that if f

is a continuous function on [a, b] then its integral

b

a

f(x) dx can be

evaluated by finding an anti-derivative. More precisely, if F (x) is an

anti-derivative of f then

b

a

f(x) dx = F (b) − F (a).

We now can present a rigorous proof of this result. We will actually

formulate the result in a slightly different way and show that the

result above follows easily from that formulation.