The Riemann Integral

30 1. The Riemann Integral
A similar proof shows that if f is continuous at b, then
∫
1 b
lim f = f(b),
h→0 + h b−h
and if f is continuous at a < c < b, then
∫
1 c+h
lim f = f(c).
h→0 + 2h c−h
More generally, if {I h : h > 0} is any collection of intervals with c ∈ I h and |I h | → 0
as h → 0 + , then
∫
1
lim f = f(c).
h→0 + |I h | I h
The assumption in Theorem 1.48 that f is continuous at the point about which we
take the averages is essential.
Example 1.49. Let f : R → R be the sign function
⎧
⎪⎨ 1 if x > 0,
f(x) = 0 if x = 0,
⎪⎩
−1 if x < 0.
Then
∫
1 h
∫
1 0
lim f(x)dx = 1, lim f(x)dx = −1,
h→0 + h 0
h→0 + h −h
and neither limit is equal to f(0). In this example, the limit of the symmetric
averages
∫
1 h
lim f(x)dx = 0
h→0 + 2h −h
is equal to f(0), but this equality doesn’t hold if we change f(0) to a nonzero value,
since the limit of the symmetric averages is still 0.
The second part of the fundamental theorem follows from this result and the
fact that the difference quotients of F are averages of f.
Theorem 1.50 (Fundamental theorem of calculus II). Suppose that f : [a,b] → R
is integrable and F : [a,b] → R is defined by
F(x) =
∫ x
a
f(t)dt.
Then F is continuous on [a,b]. Moreover, if f is continuous at a ≤ c ≤ b, then F is
differentiable at c and F ′ (c) = f(c).
Proof. First, note that Theorem 1.33 implies that f is integrable on [a,x] for every
a ≤ x ≤ b, so F is well-defined. Since f is Riemann integrable, it is bounded, and
|f| ≤ M for some M ≥ 0. It follows that
∫ x+h
|F(x+h)−F(x)| =
f(t)dt
∣ ∣ ≤ M|h|,
which shows that F is continuous on [a,b] (in fact, Lipschitz continuous).
x