I need to solve the following:
Let $f,g$ be binary function symbols, $P$ a binary predicate symbol, $c,d$ constant symbols and let $\mathcal{L} := \{f,g;P;c,d\}$.
Consider $R :=< R; +, ·; <; 0, 1 >$ as an $\mathcal{L}$-structure. Let $h$ be a unary function symbol, let $\mathcal{L}′ := \mathcal{L}∪\{h\}$ and let $\mathcal{R}′$ be $\mathcal{R}$ together with an interpretation $h_{\mathcal{R}′}$ of $h$ in $\mathcal{R}$. Find $\mathcal{L}′$-formulas $\phi$ and $\psi$ such that
i) $\mathcal{R′} \vDash \phi$ iff $h_{\mathcal{R}′}$ is continuous.
ii) $\mathcal{R′} \vDash \psi$ iff $h_{\mathcal{R}′}$ is differentiable.
I am reasonably unsure on where to start here. I need to basically figure out how to write the traditional $\forall x \in \mathbb{R}$ and $\forall \epsilon > 0$ $\exists \delta >0$ such that for $y$ such that $|x-y|< \delta$, $|f(x)-f(y)|<\epsilon$, in the language above, but I don't quite know how to formulate it. If I was shown how to do the first, I think I could attempt the differentiability one myself. Thanks for your help.