(Solved): (a) Let \( C \) be the set of continuous functions \( \mathbb{R} \rightarrow \mathbb{R} \). Define ...

(a) Let \( C \) be the set of continuous functions \( \mathbb{R} \rightarrow \mathbb{R} \). Define \( f \sim g \) on \( C \) iff \[ \exists k>0, \exists C \geq 0, \exists N \geq 0, \forall x \geq N,|f(x)-g(x)| \leq C \log (k x) \] (b) Let \( P \) be the set of non-constant polynomials \( \mathbb{R} \rightarrow \mathbb{R} \). Define \( p \sim q \) on \( S \) iff \[ \exists n \in \mathbb{N}, \exists C>0, \exists N \geq 0, \forall x \geq N,\left(\begin{array}{l} p^{(n)}(x) \not \equiv \text { constant, } q^{(n)}(x) \not \equiv \text { constant, } \\ \text { and }\left|p^{(n)}(x)-q^{(n)}(x)\right|