I am interested to know an example of a collection $S$ of real valued functions $f : \mathbb{R} \to \mathbb{R}$ with the following properties.
The set $S$ is not closed under addition. ($h(x) = f(x) + g(x)$).
The set $S$ is not closed under multiplication. ($h(x) = f(x)g(x)$).
There exists a binary operator under which the set $S$ is closed.
EDIT : This edit is being made after receiving comments and suggestions from Zev Chonoles and joriki.
The condition 3 is modified as below conditions.
3A. There exists a binary operator $\star$ under which the set $S$ is closed.
3B. The operator $\star$ on the set $S$ has associative property.
3C. $\forall f_1,f_2,f_3 \in S$ if $f_1 \star f_2 = f_3$ then $f_1 \ne f_3$ and $f_2 \ne f_3$.