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  1. MHB Craftsman

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    #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

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    #2
    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.

  4. MHB Craftsman

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    #3 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 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 ?

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    #4
    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 $f(x_1) < f(x_2)$.

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    #5 Thread Author
    Dear Deveno, thank you for your help and support Best wishes

  7. MHB Craftsman

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    #6 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 non-decreasing function ? That is, can we use "$\le$" instead of "$<$" for ${x}_{1}$ and ${x}_{2}$ ? Thank you for your attention, Best wishes

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    #7
    Quote Originally Posted by ozkan12 View Post
    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".

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