Page 181 - DMTH401_REAL ANALYSIS

Unit 13: Discontinuities and Monotonic Functions

Notes

Note If f is increasing if -f is decreasing, and visa versa. Equivalently, f is increasing if
 f(x)/f(y)  1 whenever x < y
 f(x) – f(y)  0 whenever x < y
These inequalities are often easier to use in applications, since their left sides take a very nice
and simple form. Next, we will determine what type of discontinuities monotone functions can
possibly have. The proof of the next theorem, despite its surprising result, is not too bad.

13.4 Discontinuities of Monotone Functions

If f is a monotone function on an open interval (a, b), then any discontinuity that f may have in
this interval is of the first kind.
If f is a monotone function on an interval [a, b], then f has at most countably many discontinuities.

Proof: Suppose, without loss of generality, that f is monotone increasing, and has a discontinuity
at x . Take any sequence x that converges to x from the left, i.e. x x . Then f( x ) is a monotone
0 n 0 n 0 n
increasing sequence of numbers that is bounded above by f( x ). Therefore, it must have a limit.
0
Since this is true for every sequence, the limit of f(x) as x approaches x from the left exists. The
0
same prove works for limits from the right.

Notes This proof is actually not quite correct. Can you see the mistake? Is it really true that
if x converges to x from the left then f(x ) is necessarily increasing? Can you fix the proof
n 0 n
so that it is correct?
As for the second statement, we again assume without loss of generality that f is monotone
increasing. Define, at any point c, the jump of f at x = c as:

j(c) = lim f(x) lim f(x)-
x® c + x® c -
Note that j(c) is well-defined, since both one-sided limits exist by the first part of the theorem.
Since f is increasing, the jumps j(c) are all non-negative. Note that the sum of all jumps can not
exceed the number f(b) – f(a). Now let J(n) be the set of all jumps c where j(c) is greater than 1/n,
and let J be the set of all jumps of the function in the interval [a, b]. Since the sum of jumps must
be smaller than f(b) – f(a), the set J(n) is finite for all n. But then, since the union of all sets J(n)
gives the set J, the number of jumps is a countable union of finite sets, and is thus countable.
This theorem also states that if a function wants to have a discontinuity of the second kind at a
point x = c, then it can not be monotone in any neighbourhood of c.

13.5 Discontinuities of Second Kind

If f has a discontinuity of the second kind at x = c, then f must change from increasing to
decreasing in every neighbourhood of c.
Proof: Suppose not, i.e. f has a discontinuity of the second kind at a point x = c, and there does exist
some (small) neighbourhood of c where f, say, is always decreasing. But then f is a monotone
function, and hence, by the previous theorem, can only have discontinuities of the first kind.
Since that contradicts our assumption, we have proved the corollary.

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