# Does continuity not imply existence?

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".

• 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. – Akiva Weinberger Sep 13 '15 at 6:19
• My dog can speak. Did this statement do anything to assert the existence of this dog? – Asaf Karagila Sep 13 '15 at 6:20
• @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? – Mark Viola Sep 13 '15 at 6:45
• @Dr.MV: I never bought into the whole cogitus approach. – Asaf Karagila Sep 13 '15 at 7:01
• @asafkaragila He did have a useful coordinate system ... "I think." Anyway, my attempt at humor. – Mark Viola Sep 13 '15 at 12:39