The Theorem given in the book is: A mapping $f$ of a metric space $Y$ is continuous on $X$ if and only if
$f^{-1}(V)$ is open in $X$ for every open set $V$ in $Y$
The proof given is:
I'm not sure whats happening after "Since V is open". Namely, how the $d_y$ and $d_x$ functions are created and how they relate to the delta epsilons. I figured this was using the delta epsilon definition of continuity and tried to compare it to that but couldn't come to a conclusion. Any explanation of that part of the proof would be helpful. (The second paragraph can probably be ignored for the purpose of my question; I just included it to give the whole proof)
If $V$ is open and $f(p)\in V$, since balls are basic open sets in the metric space $Y$, one finds some ball $B_{\epsilon}(f(p))$ that containing $f(p)$ such that $B_{\epsilon}(f(p))\subseteq V$. This means that for all $y\in B_{\epsilon}(f(p))$, then $y\in V$. But $y\in B_{\epsilon}(f(p))$ means that $d_{Y}(f(p),y)<\epsilon$.