# Thread: increasing, non-increasing, decreasing and non-decreasing functions

1. Please can you give definitions of increasing, non-increasing, decreasing and non-decreasing functions ? I found something but there is a lot of differents between these definitions...Can you give these definitions ? Thank you so much, Best wishes

2.

3. A (strictly) increasing function $f$ is one where $x_1 < x_2 \implies f(x_1) < f(x_2)$.

A non-decreasing function $f$ is one where $x_1 < x_2 \implies f(x_1) \leq f(x_2)$.

The dual terms are (strictly) decreasing and non-increasing (reverse the direction of the inequalities), respectively.

Most functions are none of the four, these properties are SPECIAL.

Dear Deveno,

First of all, thank you for your attention...İn some books, I saw some definitions

For example, they give these definitions as follows,

A (strictly) increasing function $f$ is one where ${x}_{1}\le{x}_{2}\implies f\left({x}_{1}\right)<f\left({x}_{2}\right)$

A non-decreasing function ${x}_{1}\le{x}_{2}\implies f\left({x}_{1}\right)\le f\left({x}_{2}\right)$

That is, they use "$\le$" instead of "<" to array ${x}_{1}$ and ${x}_{2}$...İs there any difference these definitions ?

5. Not really, the $\leq$ for the $x_1,x_2$ is unnecessary in the definition of non-decreasing, we always have for ANY function $f$:

$x_1 = x_2 \implies f(x_1) = f(x_2)$

so that does not contain any information.

$x_1 \leq x_2$ means: $x_1 = x_2$ or $x_1 < x_2$.

If $x_1 = x_2$, then $f(x_1) = f(x_2)$, so certainly $f(x_1) \leq f(x_2)$ is true (one of the two possibilities:

$f(x_1) = f(x_2)$ or $f(x_1) < f(x_2)$ is true, namely the former).

The important thing is that non-decreasing functions might have "flat spots", for example they could be constant on some interval (like step-functions corresponding to riemann sums for an increasing function).

EDIT: Using $\leq$ for a strictly increasing function leads to falsehoods: if $x_1 = x_2$, we can NEVER have $f(x_1) < f(x_2)$.

Dear Deveno, thank you for your help and support Best wishes

Dear Deveno

Also, Can we say that if " ${x}_{1}\le{x}_{2}$" $f{x}_{1}\le f{x}_{2}$ for definition of non-decreasing function ? That is, can we use "$\le$" instead of "$<$" for ${x}_{1}$ and ${x}_{2}$ ? Thank you for your attention, Best wishes

8. Originally Posted by ozkan12
Can we say that if " ${x}_{1}\le{x}_{2}$" $f{x}_{1}\le f{x}_{2}$ for definition of non-decreasing function ? That is, can we use "$\le$" instead of "$<$" for ${x}_{1}$ and ${x}_{2}$ ?
This has been answered in post #4. The properties
$x_1\le x_2\implies f(x_1)\le f(x_2)$
and
$x_1< x_2\implies f(x_1)\le f(x_2)$
are equivalent.

Also note that "non-decreasing" is not the same as "not decreasing".