Formalizing the definition of continuity and uniform continuity according to first order logic

Question

To be clear, I am familiar with the whole concept of continuity and uniform continuity, I'm just struggling with the formalization of the two statements. I get that in considering the differences of the two definitions the key is the ordering of the quantifiers but I need help none the less!

Answer 1

Let $(A,d)$ and $(B, \rho)$ be metric spaces. The function $ \ f:A \to B \ $ is continuous iff $$(\forall x \in A)(\forall \varepsilon >0)(\exists \delta>0)(\forall y \in A)[d(x,y)<\delta \to \rho \big( f(x) , f(y) \big)< \varepsilon]$$ The function $ \ f:A \to B \ $ is uniformly continuous iff $$(\forall \varepsilon >0)(\exists \delta>0)(\forall x \in A)(\forall y \in A)[d(x,y)<\delta \to \rho \big( f(x) , f(y) \big)< \varepsilon]$$ Where "$(\forall v \in X) \ P$" is an abbreviation for "$(\forall v)(v \in X \to P)$" and "$(\forall \varepsilon >0) \ Q$" is an abbreviation for "$(\forall \varepsilon)[(\varepsilon \in \mathbb{R} \wedge \varepsilon >0) \to Q]$".