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.
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