
#1
September 6th, 2015,
11:36
Please can you give definitions of increasing, nonincreasing, decreasing and nondecreasing 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

September 6th, 2015 11:36
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MHB Master
#2
September 6th, 2015,
14:28
A (strictly) increasing function $f$ is one where $x_1 < x_2 \implies f(x_1) < f(x_2)$.
A nondecreasing function $f$ is one where $x_1 < x_2 \implies f(x_1) \leq f(x_2)$.
The dual terms are (strictly) decreasing and nonincreasing (reverse the direction of the inequalities), respectively.
Most functions are none of the four, these properties are SPECIAL.

#3
September 7th, 2015,
12:53
Thread Author
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 nondecreasing 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 ?

MHB Master
#4
September 7th, 2015,
13:37
Not really, the $\leq$ for the $x_1,x_2$ is unnecessary in the definition of nondecreasing, 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 nondecreasing functions might have "flat spots", for example they could be constant on some interval (like stepfunctions 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 $f(x_1) < f(x_2)$.

#5
September 7th, 2015,
13:45
Thread Author
Dear Deveno, thank you for your help and support Best wishes

#6
September 9th, 2015,
16:20
Thread Author
Dear Deveno
Also, Can we say that if " ${x}_{1}\le{x}_{2}$" $f{x}_{1}\le f{x}_{2}$ for definition of nondecreasing function ? That is, can we use "$\le$" instead of "$<$" for ${x}_{1}$ and ${x}_{2}$ ? Thank you for your attention, Best wishes

#7
September 9th, 2015,
17:29
Originally Posted by
ozkan12
Can we say that if " ${x}_{1}\le{x}_{2}$" $f{x}_{1}\le f{x}_{2}$ for definition of nondecreasing 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 "nondecreasing" is not the same as "not decreasing".