This question already has an answer here:
So if the definition of continuity is: $\forall$ $\epsilon \gt 0$ $\exists$ $\delta \gt 0:|x-t|\lt \delta \implies |f(x)-f(t)|\lt \epsilon$. However, I get confused when I think of it this way because it's first talking about the $\epsilon$ and then it talks of the $\delta$ condition. Would it be equivalent to say: $\forall$ $\delta \gt 0$ $\exists$ $\epsilon \gt0$ $:|x-t|\lt \delta \implies|f(x)-f(t)|\lt \epsilon$. I guess what I'm asking is whether there is a certain order proofs or more formal statements need to follow. I know I only changed the place where I said there is a $\delta$ but is that permissable in a "formal" way of writing?