# Sets of Monotonic Functions

• Mar 29th 2009, 05:41 AM
woody198403
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.
• Mar 29th 2009, 08:20 AM
clic-clac
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?
• Mar 31st 2009, 02:43 AM
woody198403
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 (Angry), I think im getting things so confused (Headbang)
• Mar 31st 2009, 08:07 AM
clic-clac
(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?