ILL 1985 Problem 70

Let $C$ be a class of functions $f : \mathbb N \to \mathbb N$ that contains the functions $S(x) = x + 1$ and $E(x) = x - [\sqrt x]^2$ for every $x \in \mathbb N$ . ( $[x]$ is the integer part of $x$ .) If $C$ has the property that for every $f, g \in C, f + g, fg, f \circ g \in C$ , show that the function $\max(f(x) - g(x), 0)$ is in $C$ , for all $f; g \in C$ .