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Thread: Sets of Monotonic Functions

  1. #1
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    Sets of Monotonic Functions

    Let $\displaystyle \mathcal{M} $ be the set of montonic functions $\displaystyle \mathbb{R} \ \rightarrow \ \mathbb{R} $. That is, $\displaystyle f \ \in \ \mathcal{M}$ if either

    $\displaystyle f(x) \ \geq \ f(y) \ \forall \ x > y $,

    or

    $\displaystyle f(x) \ \leq \ f(y) \ \forall \ x > y $.

    Is $\displaystyle \mathcal{M} $ a subgroup of $\displaystyle \mathcal{F} $ under pointwise addition?

    Thank-you in advance for any help.
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  2. #2
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    Hi

    We have $\displaystyle id_{\mathbb{R}}+(-id_{\mathbb{R}})=\theta$ , where $\displaystyle \theta:\mathbb{R}\rightarrow\mathbb{R}$ is the zero map.
    Can you transform $\displaystyle id_\mathbb{R}$ or/and $\displaystyle -id_\mathbb{R}$ so their sum will become a function strictly decreasing on $\displaystyle ]-\infty ,0]$ and strictly increasing on $\displaystyle [0,+\infty[$ ? Can $\displaystyle \mathcal{M}$ be a subgroup?
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  3. #3
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    Brain Exploding

    Okay, Im really confused and in desparate need of help.

    I know that the set $\displaystyle \mathcal{F}$ of functions $\displaystyle \mathbb{R}\rightarrow\mathbb{R}$ forms a group under addition.

    Now for $\displaystyle \mathcal{M} $ to be a subgroup of $\displaystyle \mathcal{F}$ the following conditions have to be met:

    (1) For every $\displaystyle x,y, \in \mathcal{F}, xy \in \mathcal {F} $
    (2) $\displaystyle 1 \in \mathcal{F} $
    (3) For every $\displaystyle z \in \mathcal{F}, z^{-1} \in \mathcal{F} $

    But I thought that that the set $\displaystyle \mathcal{F}$ of functions $\displaystyle \mathbb{R}\rightarrow\mathbb{R}$ does not form a group under multiplication, which to me, seems to conflict with condition (1) from above.

    Ahhhhhhhhhhh , I think im getting things so confused
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  4. #4
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    (In your post, I think some occurences of $\displaystyle \mathcal{F}$ should be replaced by $\displaystyle \mathcal{M}$)

    I guess you think that $\displaystyle \mathcal{M}$ doesn't satisfy (1), and it is true.

    I was trying to give you something easy to visualize in my last post.
    For example, let $\displaystyle f$ and $\displaystyle g$ be the functions defined by:

    $\displaystyle f(x)=\begin{cases}x & x\leq 0 \\ 2x & x\geq 0\\\end{cases}$

    $\displaystyle g(x)=\begin{cases}-2x & x\leq 0 \\ -x & x\geq 0 \\\end{cases}$

    Are they in $\displaystyle \mathcal{M}$ ? And their pointwise addition?
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