372 Chapter 7 • The Riemann Integral
- Prove that Theorem 7.2.14 remains true if "< c" is replaced by ":S c".
- Prove Theorem 7.2.15. [Hint: Apply Theorem 7.1.5 to Riemann's criterion
and the sets A and B of Definition 7.2.6.]
19. Prove Case 2 of Theorem 7.2.16.- Define the function f : [O, 1] -t JR by f(O) = 0 and f(x) = ~ if n~l <
x < ~; for n E N. Sketch the graph of f and prove that f is Riemann
integrable on [O, l]. Notice that f has discontinuities at infinitely many
points in [O, l]. - Suppose Q is a refinement of the partition P of [a, b] containing just one
point not in P. Let P = {xo, x1, x2, · · · , Xn}, Q = {xo, X1, X2, · · · , Xk-1, x;,,
Xk, ... ,xn}, mk = inff[xk-1,xk], mk,1 = inff[xk-1,x;,] and mk,2 =
inf f[x;,, xk]· Prove that S(f, Q)-S(f, P) = mk,1 (x;,-xk-1) +mk,2(xk -
x;,) - mk(Xk - Xk- 1 ). (See the proof of (a) of Theorem 7.2.4.) State and
prove a similar formula for S(f, Q) - S(f, P).
7 .3 The Integral as a Limit of Riemann Sums
It seems that every important concept in calculus involves the notion of limit.
The integral is no exception, as we shall now see. The t ype of limit Riemann
developed for this purpose is different from any of the kinds of limits we have
seen so far. Yet it will still "feel" like a limit- it will be expressed with c and
J playing familiar roles. We begin with some technical preliminaries.
Definition 7.3.1 The mesh of a partition P of [a, b] is the length of the longest
subinterval [xi-l, xi] between consecutive points of the partition P; in symbols,llPll =max{ xi - Xi-1 : 1 :Si :Sn}.Having defined the mesh of a partition, we shall immediately put this con-
cept to work.Theorem 7.3.2 (Riemann/Darboux Criterion for Integrability)
A bounded function f :[a, b] -t JR is integrable over [a, b] -¢:::==}
lim (S(f, P) - S_(f, P)) = 0, in the sense that
/IPll-->O/ vc > 0, :l J > 0 3 I:/ partitions P of [a, b], llPll < J =? S(f, P) - S_(f, P) < c. /
[Equivalently, there is some k > 0 such that Ve > 0, :l J > 0 3 I:/ partitions P
of [a, b], llPll < J =? S(f, P) - S_(f, P) <kc.]