The Riemann Integral

26 1. The Riemann Integral
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0 0.05 0.1 0.15 0.2 0.25 0.3
Figure 4. Graph of the Riemann integrable function y = sgn(sin(1/x)) in Example 1.44.
where sgn is the sign function,
⎧
⎪⎨ 1 if x > 0,
sgnx = 0 if x = 0,
⎪⎩
−1 if x < 0.
Then f oscillates between 1 and −1 a countably infinite number of times as x →
0 + (see Figure 4). It has jump discontinuities at x = 1/(nπ) and an essential
discontinuityatx = 0. Nevertheless,itisRiemannintegrable. Toseethis, notethat
f is bounded on [0,1] and piecewise continuous with finitely many discontinuities
on [r,1] for every 0 < r < 1. Theorem 1.42 implies that f is Riemann integrable
on [r,1], and then Theorem 1.39 implies that f is integrable on [0,1].
1.8. The fundamental theorem of calculus
In the integral calculus I find much less interesting the parts that involve
only substitutions, transformations, and the like, in short, the parts that
involve the known skillfully applied mechanics of reducing integrals to
algebraic, logarithmic, and circular functions, than I find the careful and
profound study of transcendental functions that cannot be reduced to
these functions. (Gauss, 1808)
The fundamental theorem of calculus states that differentiation and integration
are inverse operations in an appropriately understood sense. The theorem has two
parts: in one direction, it says roughly that the integral of the derivative is the
original function; in the other direction, it says that the derivative of the integral
is the original function.