292 Chapter 5 • Continuous FunctionsOSCILLATION OF A FUNCTION
Before learning more about the sets of discontinuity of functions, we make
what may seem like only an interesting detour. You will eventually see the
purpose of this detour. We define the oscillation of a function at a point
as a measure of how wildly discontinuous the function is at that point. We first
define the oscillation of a function on a set, and then use that definition to
define the oscillation of the function at a specific point.Definition 5.7.4 Let f : D(f) ----+ JR, A ~ D(f). If f is bounded on A, we
define the oscillation of f on A to be W1(A) =sup f(A) - inf f(A). If f is
unbounded on A , we define W1(A) = +oo.Exercise 5.7.5 Prove that if A~ B, then W1(A) :::=; W1(B).Definition 5. 7.6 Let x 0 E D(f). Define the function WJ,xo (0, +oo) ----+
JR u { + 00} by WJ,xo(c) = W1 (N,,(xo) n D(f)).
Exercis e 5.7.7 Prove that WJ,xo: (0, +oo)----+ JR U { +oo} is monotone increas-
ing.Exercise 5. 7 .8 Prove that 'ixo E D(f), lim w f ,xo ( c) exists in JR U { + oo}.
o--+O+Definition 5.7.9 For each x 0 E D(f), define the oscillation of f at x 0 to
be w1(xo) = lim WJ,x 0 (c).
o--+O+
The function WJ is often called the saltus function, and w1(x 0 ) is often
called the saltus of f at xo.
Exercise 5.7.10 Prove that w1(xo) :2: 0, and f is continuous at xo iff w1(xo) =
0.
Theorem 5. 7. 11 The set of discontinuities of a function f : D(f) ----+ JR is the
union of a countable collection of closed sets.
Proof. Proceed as follows. Suppose f : D(f) ----+ R
(a) Ve> 0, let S,,(f) = {x E D(f): w1(x) :2: c}. Prove that S,,(f) is closed.
[See Exercise 5.1.23.]
(b) Prove that 81 (f) ~ S 1 2 (!) ~ S 1 3 (!) ~ .. · ~ Si n (f) ~ .. ., and t hat
00
LJ Si (f) is the set of points where f is discontinuous. •
n=l n