Monotonic functions

Note

Let $I\subseteq \mathbb{R}$ be an interval. A function $f:I\to \mathbb{R}$ is called monotonic non-decreasing if $f(x)\le f(y)$ for every $x,y\in I$ such that $x\le y$. When the same holds with the reversed inequalities is called monotonic non-increasing.

Properties §

Let $f:I\to \mathbb{R}$ be a monotonic function. Then

1. $f$ is measurable.

Note

$f^{-1}((-\infty ,a])$ is an interval, since $(-\infty ,a]$ is a Pi-system that generates the Borel $\sigma$-algebra, $f$ is measurable. $\square$

1. For every $x\in \mathring{I}$ the left and right limit exists (they are finite).

Note

WLOG let $f$ be non-decreasing, let $\{x_n\}$ be a monotoinic sequence >such that $x_n\uparrow x$, then

$$\underset{y\uparrow x}{\lim }f(y)=\underset{n\to \infty }{\lim }f(x_n)\le f(x)$$

>exists and is finite, since the sequence $f(x_n)$ is non-decreasing and

>bounded from above by $f(x)$. Similarly one gets the result for non
>increasing and right limits. $\square$

Note

If we don’t restrict to point in the interior, we have existence but not necessarily finiteness (we can’t bound the sequence), consider for example $I=(-\frac{\pi }{2},+\frac{\pi }{2})$ and the funtion $\tan (x)$.

3. Every discountinuity is of jump type, and the number of discontinus points is at most countable.

Note

Let $[a,b]\subseteq I$ a finite interval. Call $S$ the set of >discontinuities of $f$ in $[a,b]$. This means that there exists an $n>0$ >such that $|f_{-}(x)-f_{+}(x)|>\frac{1}{n}$ . Then we can write $S$ as >the union >$$ S = \bigcup_{n =1}^\infty \left{x \in [a,b] ,:, |f_-(x)- f_+(x)| > \frac 1 n\right}

>since the interval is finite, each set in the union is finite, since has at >most $|f(b)-f(a)|n$ elements. Then $S$ is at most numerable. >Since $\mathbb{R}$ is can be writte has the countable union of finite

>intervals, the number of discontinuities is still at most countable. $\square$

The last property implies that a monotonic function is continuous almost everywhere. It turns out that this also holds for its derivative.

Note

Every monotonic non-decreasing function $f:[a,b]\to \mathbb{R}$ admits a derivative a.e. Furthermore $f^{\prime }$ satisfies the inequality:

$${\int }_a^b{f}^{\prime }(t)dt\le f(b)-f(a)$$

Note

First we prove that the set

$$E:=\{x\in (a,b):{\underline{D}}^{+}f(x)<\alpha <\beta <{\overline{D}}_{-}f(x)\}$$

has measure zero. \vdots \vdots \vdots Define the familiy of functions $\{{\Delta }_{h}f{\}}_{h\ge 0}$ where

$${\Delta }_{h}f(x):=\frac{f(x+h)-f(x)}{h}$$

(we do a continuous extention of $f$ outside $[a,b]$). Clearly ${\Delta }_{h}f\to f^{\prime }$ a. e., using Fatou’s lemma:

$${\int }_a^b{f}^{\prime }dx\le \underset{h\to 0}{\mathrm{liminf}}{\int }_a^b{\Delta }_{h}fdx=\underset{h\to 0}{\mathrm{liminf}}\frac{1}{h}{\int }_b^{b+h}fdx-\frac{1}{h}{\int }_a^{a+h}fdx$$ $$\le \underset{h\to 0}{\mathrm{liminf}}\frac{1}{h}{\int }_b^{b+h}f(b)dx-\frac{1}{h}{\int }_a^{a+h}f(a)dx=f(b)-f(a)\square$$

Note

Every left-continuous monotonic function $f$ can be written as the sum of a continuous monotonic function $\varphi$ and a jump function $H$ (left-continuous). $f=\varphi +H$ This representation is unique.

Note \{x_n\} and \{h_n\} be the set of discontinuities and jump values of f, define the jump function H with these values. The function \phi := f - H is continuos (easy to check) and monotonic. Suppose WLOG that f is non-decreasing, then for x \leq y

Let

$$\varphi (y)-\varphi (x)=f(y)-f(x)-[H(y)-H(x)]\ge 0$$

since the sum of the jumps from $x$ to $y$ is not greater the the whole increment of the function. $\square$

Clearly we can generalize this decomposition to monotonic function that have bot left and right continuous point, by summing two Jump functions.