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I am reading these lecture notes and there is a converse to the Cauchy Riemann equations and it is stated as follows:

If the partial derivatives exist and are continuous then $f$ is (complex) differentiable.

My question is: Doesn't continuity imply "exist"? Meaning, we could just say if the partial derivatives are continuous then $f$ is differentiable? Or what does "exist" mean in this context? I assumed it meant "is finite".

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  • $\begingroup$ Well, perhaps a student will prove that the derivative of some function is continuous at each point of the derivative's domain, but forget to prove that the domain is all of $\Bbb C$. So they want to stress that the function has to exist everywhere. $\endgroup$ – Akiva Weinberger Sep 13 '15 at 6:19
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    $\begingroup$ My dog can speak. Did this statement do anything to assert the existence of this dog? $\endgroup$ – Asaf Karagila Sep 13 '15 at 6:20
  • $\begingroup$ @AsafKaragila Didn't a famous mathematician once say, "Your dog can speak. Therefore, it is."? I would simply state ... Your dog can speak? How many languages? $\endgroup$ – Mark Viola Sep 13 '15 at 6:45
  • $\begingroup$ @Dr.MV: I never bought into the whole cogitus approach. $\endgroup$ – Asaf Karagila Sep 13 '15 at 7:01
  • $\begingroup$ @asafkaragila He did have a useful coordinate system ... "I think." Anyway, my attempt at humor. $\endgroup$ – Mark Viola Sep 13 '15 at 12:39
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You're getting caught up in something that has nothing to do with math.

I would read "partial derivatives are continuous" and "partial derivatives exist and are continuous" the same exact way. The latter is just more explicit. Either way, this really is a minor thing that has little to do with math.

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    $\begingroup$ @GitGud No worries. $\endgroup$ – user223391 Sep 13 '15 at 6:34
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    $\begingroup$ However, while it's true that it is a minor thing that he's getting caught up on, paying close attention to the details and subtle nuances of statements is very important. $\endgroup$ – Shane T Sep 13 '15 at 6:45

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