14 1. The Regulated and Riemann Integrals
Theorem 1.6.1. If f is a continuous function and we define
F (x) =
x
a
f(t) dt,
then F is a differentiable function and F (x) = f(x).
Proof. By definition
F (x0) = lim
h→0
F (x0 + h) F (x0)
h
;
so we need to show that
lim
h→0
F (x0 + h) F (x0)
h
= f(x0),
or, equivalently,
lim
h→0
F (x0 + h) F (x0)
h
f(x0) = 0.
To do this we note that
F (x0 + h) F (x0)
h
f(x0) =
x0+h
x0
f(t) dt
h
f(x0) (1.6.1)
=
x0+h
x0
f(t) dt f(x0)h
h
=
x0+h(f
x0
(t) f(x0)) dt
|h|
.
Proposition 1.2.1 tells us that
x0+h
x0
(f(t) f(x0)) dt
x0+h
x0
|f(t) f(x0)| dt
Combining this with equation (1.6.1) above we obtain
(1.6.2)
F (x0 + h) F (x0)
h
f(x0)
x0+h
x0
|f(t) f(x0)| dt
|h|
.
But the continuity of f implies that given x0 and any ε 0 there
exists δ 0 such that whenever |t−x0| δ we have |f(t)−f(x0)| ε.
Thus, if |h| δ, then |f(t)−f(x0)| ε for all t between x0 and x0 +h.
It follows that
x0+h
x0
|f(t) f(x0)| dt ε|h|
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