A function that lies between the ordered sets which are either preserving or reversing the order given to us is known as a monotone function or a monotonic function.

Definition

If $f$ is a function which is defined on an interval say I, then, we according to monotonicity we have,

1) For every $a$, $b$ in $I$ such that $a\ <\ b$, $f$ is increasing will imply that $f\ (a)\ <\ f\ (b)$.

2) For every $a$, $b$ in $I$ such that $a\ <\ b$, $f$ is decreasing will imply that $f\ (a)\ >\ f\ (b)$.

3) If $f$ is either increasing or decreasing on an interval $I$, then we say $f$ is strictly monotonic on $I$.

As per the rule that it is increasing we call the function to be monotonically increasing and if the function is decreasing we call it monotonically decreasing.

Monotonicity Theorem

According to the monotonicity theorem if $g$ is a continuous function on interval $I$ and also $g$ is differentiable in $I$ everywhere, then,

1) If for all $x$in an interval, $g’\ (x)\ >\ 0$, then $g$ is increasing in that interval.

2) If for all $x$ in an interval, $g’(x)\ <\ 0$, then $g$ is decreasing in that interval.

Monotonicity Applications and Results

If $g$ is a monotonic function of real number set as domain and range, then the following are true:

1) The function $g$ will always have limits from right and left as well at every domain point.

2) The limits of $g$ can be positive infinity, negative infinity or a real number at positive or negative infinity.

3) The function here can have jump discontinuities only.

4) The function can have only countable discontinuities in the domain of it.

5) When $g$ is defined on $I$ monotonically, then $g$ is differentiable on every point in $I$.

6) When $g$ is a defined on the interval [$a$, $b$] monotonically, then $g$ in Reimann integrable type function.

Monotonicity Function

When we have a topological vector space say $Z$, then an operator $*$, such that $T\ :\ Y\ \rightarrow\ Y*$ is called a monotone operator only if we have,
$(T\ a\ –\ T\ b$, $a\ –\ b$) $\geq$ $0$, for all $a$, $b$ $\in$ $Y$

Also, a subset $H$ of $Y\ \times\ Y*$ is known as a monotone subset only if for each pair [$a1$, $b1$] and

[$a2$, $b2$] in H we have,
($b1\ –\ b2$, $a1\ –\ a2$) $\geq$ $0$

$H$ is known as maximal monotone if and only if H is maximal in the sense of set inclusion property among all monotone sets. Also, the graph of the defined monotone operator

$H\ (T)$ will now be a monotone set. Again, we call the monotone operator as maximal operator if and only if the graph of this operator is a maximal monotone set.

Monotonicity Theory

The definition of monotonicity is valid in the cases of partially ordered sets as well as preordered sets along with real numbers. We avoid the terms decreasing, increasing in this case as they give a conventional representation in a pictorial form which is not applicable to the not total orders.

An isotone is a monotone function, also called order preserving. The antitone is the dual notion, also known as order reversing or anti monotone. This implies that an order reversing function will satisfy the following property
$a\ \leq\ b$

$\Rightarrow$ $g\ (a)\ \geq\ g\ (b)$

for every $a$, $b$ lying in the domain of the function. It is important to note that the composition of two monotone mappings is again a monotone.

When we have a constant function, it lies in both categories, as it is both antitone and monotone. This condition is a vice versa situation.

Examples

Let us look at a couple of examples to understand the criteria of monotonicity better.

Example 1:

In the interval [$0$, $2\ \pi$], $h\ (y)$ = $cos^2\ y$, is increasing or decreasing in which subset?

Solution:

We have $h\ (y)$ = $cos^2\ y$

$\Rightarrow$ $h’(y)$ = $-\ 2\ cos\ y\ sin\ y$

$-\ 2\ cos\ y\ sin\ y$ = $0$

$\Rightarrow$ $-\ sin\ (2\ y)$ = $0$

$\Rightarrow$ $2\ y$ = $0,\ \pi,\ 2\pi,\ 3\pi,\ 4\pi$

$\Rightarrow$ $y$ = $0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\ \pi}{2}$, $2 \pi$

We take point $\frac{\pi}{4}$, $\frac{3\ \pi}{4}$, $\frac{5\ \pi}{4}$, $\frac{7 \pi}{4}$ from each interval in between points above and see that, $h$ is increasing in interval

($\frac{\pi}{2}$, $\pi$) $\cup$ ($\frac{3 \pi}{2}$, 2$\pi$) and $h$ is decreasing in interval

($0$, $\frac{\pi}{2}$) $\cup$ ($\pi$, $\frac{3 \pi}{2}$).
Example 2:

Find if $g$ is increasing or decreasing when $g\ (x)$ = $x^3\ –\ 3\ x^2\ +\ 12$

Solution:

Here $g\ (x)$ = $x^3\ –\ 3\ x^2\ +\ 12$

$\Rightarrow$ $g’(x)$ = $3\ x^2\ –\ 6x$

$\Rightarrow$ $g’(x)$ = $3\ x\ (x\ –\ 2)$

$g’(x)$ = $0$

$\Rightarrow$ $3\ x\ (x\ –\ 2)$ = $0$

$\Rightarrow$ $x$ = $0,\ 2$

When we pick values from each interval obtained from above say, $-1$, $1$, $3$, we find that, function $g$ is increasing in ($2$,$\infty$)and decreasing in interval ($- \infty$, $0$) $\cup$ ($0$, $2$)