Do functions where smaller $V$ correlates to larger $f^{-1}(V)$ exist?

Question

I've been studying $\epsilon - \delta$ continuity and, more generally, continuity at a point using open sets. I know it's possible that for some open set $V$ of $f(x)$, the pre-image $f^{-1}(V)$ can get smaller as $V$ gets smaller. I also know $f^{-1}(V)$ may remain of same size as $V$ gets smaller. I'm concerned about situations where as $V$ gets smaller, $f^{-1}(V)$ gets larger. Is this possible?

Answer 1

It is impossible for $V$ to get smaller and $f^{-1}(V)$ to become larger. By definition $$f^{-1}(V) = \{ x : f(x) \in V\}$$

Now if we take some $$V' \subseteq V$$ then its inverse is $f^{-1}(V') = \{ x : f(x) \in V'\}$. But if $x \in f^{-1}(V')$, then $f(x) \in V' \subseteq V$, which means $x \in f^{-1}(V)$. So $$f^{-1}(V') \subseteq f^{-1}(V)$$

So as $V$ decreases in size, at best $f^{-1}(V)$ can stay the same size. It cannot get larger.