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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.

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    I hope I'm reading it correctly, though I think you're just saying that given e > 0 there exists a d > 0 s.t. f(B(d,x)) is a subset of B(e,f(x)) ? Right? The preimage definition has tremendous importance as it generalizes the concept of nhoods to that of cts functions preserving open sets.2012-01-04
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    A mathematician, writing an argument that requires a proof of continuity, will sometimes use that characterization. And (except at the student level) there would be no special comment that this is equivalent to some other definition.2012-01-04
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    Dear kjo: A topology can be defined by specifying all the open sets, or the neighborhoods of each point, or the closure of each subset. I'd expect your definition to be in Bourbaki. Indeed, the notion of **filter** is one of their specialities. I don't see what's nonstandard in your definition. Of course, your point is an outstanding one!2012-01-04
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    You make an interesting point about *filters*, @Pierre-Yves. Can you elaborate the connection to them in an answer?2012-01-04

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