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.