Principles of Mathematics in Operations Research

11.5 Monotonic Functions 165

f(x-)

f(x)

f(x+)

f

^0

Fig. 11.4. Proof of Theorem 11.5.3

Corollary 11.5.4 Monotonic functions have no discontinuities of the second

type.

Theorem 11.5.5 Let f : (a, b) H» K be monotonic. Let A be the set of dis-

continuous points of f, then A is at most countable.

Proof. Assume / is decreasing, then A = {x G {a,b} : f(x+) < f(x—)}. Vx G

A, find f(x) G Q B f(x+) < r(x) < f(x-) and fix r(x). Define g : A i-» <Q> by

g(x) = r{x). We will show that g is one-to-one: Let x\ ^ X2 G A,X\ < X2 =>-

r(xi) > f(xi+) > f{x2—) > r(x2) => r(x\) / r(x2). Thus, <? is one-to-one,

and A is numerically equivalent to Q by g(x) = r(x). Therefore, ^4 is at most

countable. •

Remark 11.5.6 The points in A may not be isolated. In fact, given any
countable subset E of (a, b) (E may even be dense), there is a monotonic
function f : (a,b) H-> R B f is discontinuous at every x G E and continuous
at every other point. The elements of E as a sequence {xi,£2,...}. Let cn >
0 3 Ylcn is convergent. Then, every rearrangement Yl c4>(n) a^so converges
and has the same sum. Given x G (a, b) let Nx = {n : xn < x}. This set may
be empty or not. Define f(x) as follows

1 luneNx c»> otherwise

This function is called saltus function or pure jump function.

(a) f is monotonically increasing on (a,b):
Leta<x<y<b.IfNx=0, f(x) = 0 and f(y) > 0.
// Nx jL 0, x < y =» f(x) = Enejv. c» < T.neNy ^ = f(y).
(b) f is discontinuous at every xm G E:
Letxm G E befixed. f{xm+) = MXm<t<bf(t), f{xm-) = supa<s<Xm f(s).
Let xm < t < b, a < s < xm be arbitrary =>• a < s < xm <t<b. Then,
NscNt, m<E;Nt, m<£Ns=>meNt\Ns.
J\t) ~ j(S) — Z^nENt C" — 2-m€NB C« ~ Z^nENa\Nt C» — Cm =>