Consider this definitions:
A function $f:X \to Y$ is continuous at $x\in X$ iff for any open neighborhood $V_{f(x)}$ of $f(x)$ there is an open neighborhood $U_{x}$ of $x$ that gets mapped by $f$ into $V_{f(x)}$ (or, in other words, there is an open neighborhood $U_{x}$ of $x$ such that $f[U_x]\subseteq V_{f(x)}$). A function $f:X \to Y$ is continuous iff it is continuous at all $x \in X$.
This definition of continuity seems to me equivalent to the "standard" definition in terms of inverse images (namely $f:X\to Y\;$ is continuous iff for any open set $V\subseteq Y$, the set $f^{-1}(V)\subseteq X$ is open).
Am I wrong?
Assuming that I'm correct, I am baffled by the prevalence of the currently standard definition, since the one above looks to me far more natural. It is certainly more obviously a generalization of the $\epsilon$-$\delta$ definition of continuity in metric spaces (just replace $V_{f(x)}$ and $U_{x}$ by open balls $B(f(x), \epsilon)$ and $B(x, \delta)$, respectively), which, in turn, is an obvious generalization of the $\epsilon$-$\delta$ definition of continuity for functions $\mathbb{R}\to\mathbb{R}$ (just replace the open balls $B(f(x), \epsilon)$ and $B(x, \delta)$ by open intervals $(f(x) - \epsilon, f(x) + \epsilon)$ and $(x-\delta, x+\delta)$, respectively).
Given these considerations, why is the standard definition the generally preferred one?
Edit: the replies I've gotten so far have focused on the fact that the alternative definition depends on an auxiliary definition of continuity at a point, but this is a very minor aspect of the alternative definition. I chose this two-part approach only to make the wording of the definition of continuity slightly less awkward, but it is not required. I could have just as well written:
A function $f:X \to Y$ is continuous iff for all $x \in X$ and any open neighborhood $V_{f(x)}$ of $f(x)$ there is an open neighborhood $U_{x}$ of $x$ such that $f[U_{x}]\subseteq V_{f(x)}$.
Also, these replies suggest that, when it comes to defining terms, brevity trumps clarity. I find this hard to take: a definition, by definition, is introducing a concept, so its intended audience is one that will appreciate clarity over brevity. An equivalent characterization of the same concept whose only advantage is greater brevity should be relegated to a theorem, IMO.