(adsbygoogle = window.adsbygoogle || []).push({}); Complex differentiable <--> real and imaginary parts satisfy C-R eqns and are cont.

Say we have a complex function f(z) we can break this in to real and imaginary parts:

f(z)=u(x,y)+iv(x,y)

In my book I am told the following:

(1) f complex differentiable at z_{0}in ℂ --> the Cauchy Reimann equations are satisfied by the partials of u and v at z_{0}

I am also told seperately:

(2) For some z_{0}in ℂ, if the partials of u and v exist and are continuous on an open set containing z_{0}and they satisfy the Cauchy Reimann equatiions, then f is complex differentiable at z_{0}

Why isn't this an iff thing? It seems like the only thing keeping it from being bidiriectional is the fact that in (2) I also have to check that the partials are continuous on an open set O containing z_{0}. But this is where my doubt arises. I am also told in my book:

(3) If u and v satisfy the Cauchy Reimann equations at z_{0}for some neighborhood O containingi z_{0}, then u and v are both harmonic in O.

Since being harmonic requires establishing an equality between the second partial derivatives, isn't such a thing only possible if the first partial derivatives are continuous (as continuity is a essential for differentiability)? So how if u and v are harmonic shouldn't their first partials be continuous? If (1) gaurantees the Cauchy Reimann equations are satisfied which allows me to invoke (3), then doesn't (1) also gaurantee that the partials are continuous?

So why can't I simply say f is differentiable at z_{0}in ℂ ⇔ it's real and imaginary components have partial derivatives continuous on an open set O containing z_{0}and these partials satisfy the Cauchy Reimann equations. Why does this need to be broken in to two separate theorems?

SIDE QUESTION: I know that if a complex function f is differentiable at z_{0}it is actually infinitely differentiable there. But are it's real and imaginary parts also infinitely differentiable? I am just getting so confused separating what I can infer about the complex function and what I can infer about its real and imaginary parts which are functions of real variables.

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# Complex differentiable <-> real and imaginary parts satisfy C-R eqns and are cont.

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