MHB Increasing, non-increasing, decreasing and non-decreasing functions

[ozkan12]

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 :)

 

[Deveno]

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.

 

[ozkan12]

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 ?

 

[Deveno]

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 $f(x_1) < f(x_2)$.

 

[ozkan12]

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

 

[ozkan12]

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 :)

 

[Evgeny.Makarov]

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".