Page 180 - DMTH401_REAL ANALYSIS

Real Analysis

Notes This function is impossible to graph. The picture above is only a poor representation of the true
graph. Nonetheless, take an arbitrary point x on the real axis. We can find a sequence {x } of
0 n
rational points that converge to x from the right. Then g(x ) converges to 1. But we can also find
0 n
a sequence {x } of irrational points converging to x from the right. In that case g(x ) converges
n 0 n
to 0. But that means that the limit of g(x) as x approaches x from the right does not exist. The
0
same argument, of course, works to show that the limit of g(x) as x approaches x from the left
0
does not exist. Hence, x is an essential discontinuity for g(x).
0
It is clear that any function is either continuous at any given point in its domain, or it has a
discontinuity of one of the above three kinds. It is also clear that removable discontinuities are
‘fake’ ones, since one only has to define f(c) = limf(x) and the function will be continuous at c.
x® c
Of the other two types of discontinuities, the one of second kind is hard. Fortunately, however,
discontinuities of second kind are rare, as the following results will indicate.
13.3 Monotone Function

A function f is monotone increasing on (a, b) if f(x)  f(y) whenever x < y. A function f is
monotone decreasing on (a, b) if f(x)  f(y) whenever x < y.
A function f is called monotone on (a, b) if it is either always monotone increasing or monotone
decreasing.
Some basic applications and results

The following properties are true for a monotonic function f : R ® R:
 f has limits from the right and from the left at every point of its domain;

 f has a limit at infinity (either  or –) of either a real number, , or –.
 f can only have jump discontinuities;

 f can only have countably many discontinuities in its domain.
These properties are the reason why monotonic functions are useful in technical work in analysis.
Two facts about these functions are:
 if f is a monotonic function defined on an interval I, then f is differentiable almost
everywhere on I, i.e. the set of numbers x in I such that f is not differentiable in x has
Lebesgue measure zero.

 if f is a monotonic function defined on an interval [a, b], then f is Riemann integrable.

An important application of monotonic functions is in probability theory. If X is a random
variable, its cumulative distribution function

F (x) = Prob (X  x)
X
is a monotonically increasing function.

A function is unimodal if it is monotonically increasing up to some point (the mode) and then
monotonically decreasing.

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