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In every textbook the condition for a function $f$ to be continuous at a point $x_0$ are:
Are conditions 1 and 2 implicit in 3? Or would a logicist get mad if books didn't include them?
I agree with you that the default assumption of $A = B$ should be that both $A$ and $B$ exist. But it couldn't hurt to be explicit about it, and this is one of the most important (and initially confusing!) definitions in all of calculus.
If you think about it, the symbol $\lim_{x \rightarrow x_0} f(x)$ is weird, because it denotes something that may or may not exist! So yes, I can well see that a logicist would find some problems there.
(The way I usually express the definition is "The limit at $x_0$ exists and is equal to the value of the function at $x_0$"; I want to remind people that the limit might not exist, since in practice this is the issue most of the time.)
Note that the related symbol $\sum_{n=1}^{\infty} a_n$ is even worse: this is simultaneously used to denote both the infinite series (i.e., the sequence of partial sums $S_n = a_1 + \ldots + a_n$) and its limit, if it exists. It is interesting to reflect on this point: once you see it explicitly, you think that this type of overloaded usage of a symbol should be dangerous and lead to vast confusion...except that it seems not to, in practice. Notation is remarkably durable: the common phrase "abuse of notation" usually refers to bad notation that you can get away with.
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Afterthought: at least once you get to the point where you are seeing formal -- i.e., $\epsilon$--$\delta$ -- definitions, I think that this definition has things the more complicated way around. That is, rather than talking about limits first and then continuity, why don't we define continuity at $x_0$ first -- that's a simpler definition, since we don't need to use deleted intervals -- and then define $\lim_{x \rightarrow x_0} f(x)$ to mean that we can (re)define $f$ at $x_0$ so as to make the function continuous at $x_0$.
This is even useful at the informal, freshman calculus level: it exploits the idea that students have some intuition for what it means for a function to be continuous at a point -- roughly, it should mean that it is a nice, unbroken curve at that point. Of course that's not exactly right or totally precise, but it's pretty good for a first calculus course, and it stands in contrast to the notion of a limit, in which the "we giveth / we taketh away" game with the point in question is so vexing as to prevent many students from seeing any intuition behind it whatsoever.
Yes, of course they are, but introductory texts do this so as to "lay it our cleanly" so to speak. Exercises of these sort are posed slightly as a "checklist": first see if $f$ is defined at $x_0$. Is it? Good. Let's see if it has a limit. Does it? Great. Now, do the previous quantities match? Awesome. Your function is continuous there.
