4.1.3 The Riemann Integral
The reader is assumed to be familiar with the Riemann integral, but if not, the above is more general and
gives the principal results for the Riemann integral of continuous functions very easily. Therefore, here is a
slight digression to show this.
In the special case that f :
, and γ
the Rieman Stieltjes sums reduce to ordinary
and the above argument reduces to showing the existence of the Riemann integral ∫
continuous function. Note that it is obvious that if f
0 for all t,
which together imply that
the triangle inequality.
As to vector valued functions and Riemann integration, if f :
is continuous, what is meant
? We mean the obvious thing:
Now consider the following sequence of steps using the Cauchy Schwarz inequality, Proposition 1.10.1, from
the first to the second line.
Divide by the first term on the right. This gives
which holds even if ∑
= 0. This proves the following convenient version of the triangle
inequality, this time for vector valued functions.
Proposition 4.1.5 Let f :
→ ℝp be continuous. Then
Also one obtains easily the fundamental theorem of calculus.
Theorem 4.1.6 Let f :
→ ℝ be continuous and real valued. Suppose F′
for all x ∈
where F is a continuous function on
Proof: Let ε > 0. There is a δ > 0 such that if
then for P
By intermediate value theorem from calculus, there is τi ∈
. Since ε >
it follows that ∫
In case of vector valued continuous functions with f
The other form of the fundamental theorem of calculus is also obtained.
Theorem 4.1.7 Let f :
→ ℝp be continuous and let F
ds. Then F′
for all t ∈
Proof: Let t ∈
be small enough that everything of interest is in
. First suppose
provided that ε is small enough due to continuity of f at t. A similar inequality is obtained if h < 0 except
in the argument, you will have t + h < t so you have to switch the order of integration in going to the
second line and replace 1∕h with 1∕