## 6.3. Discontinuous Functions

### Theorem 6.3.6: Discontinuities of Monotone Functions

*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 _{0}*.
Take any sequence

*x*that converges to

_{n}*x*from the left, i.e.

_{0}*x*. Then

_{n}x_{0}*f( x*is a monotone increasing sequence of numbers that is bounded above by

_{n})*f(x*. Therefore, it must have a limit. Since this is true for every sequence, the limit of

_{0})*f(x)*as

*x*approaches

*x*from the left exists. The same prove works for limits from the right.

_{0}**Note:** This proof is actually not quite correct. Can you
see the mistake ? Is it really true that if *x _{n}*
converges to

*x*from the left then

_{0}*f(x*is necessarily increasing ? Can you fix the proof so that it is correct ?

_{n})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) = f(x) - f(x)

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.