Is it then correct to say that if the Cauchy-Riemann equations are satisfied in $z_0 \in \mathbb C$, then $f'(z_0)$ exists ? I was wondering, because the theorem about these equations says always let $f: U \rightarrow \mathbb C$ with $U$ open subset in $\mathbb C$.

Yeah it exists there. You have $f:\mathbb{C}\to \mathbb{C}$ so you surely do have an open subset

According to proofwiki.org/wiki/Cauchy-Riemann_Equations I must have that the equations are satisfied in the whole domain of $f$. But how can I then check what the (isolated) points are where $f'$ exists ?

Oh as far as I can see they don't distinguish between complex differentiable and holormophic. That may be because being complex differentiable in only one point is not that interessting. As you will see complex differentiable on an open subset is a very strong thing

Just for clarification: In order to compute ALL points $z$ where $f'(z)$ exists (thus complex differentiable is) it is sufficient to check in which points $z$ the Cauchy-Riemann equations are satisfied ?

We defined complex differentiable via real differentiable + satisfying cauchy riemann, How did you define it ?

Let $f : \Omega \rightarrow \mathbb C$ where $\Omega$ is a region (open and connected). If $$ \lim_{h \rightarrow 0, h \in \mathbb C \backslash \{0 \}} \frac{ f(z_0 + h) - f(z_0) } h $$ exists then $f'(z_0)$ is this limit and we say $f$ is holomorphic at the point $z_0$.

your proof only cares about being complex differnetiable in the whole subset

maybe i wasnt clear about that but my task was to show in which points the functions as a complex derivative.

oh ok I see we did call things different in my lecture.

But the proof works even if you work pointwise

I am quite confused about the CR equations, because the theorem on the wiki page makes no statements about single point, but only about open subsets where on which $f$ is then holomorphic

oh ok

you will later see why it is so

because being only complex differentiable at a single point is not helpfull, but when you have a open subset and know it is 1 time differentiable you even know it its arbitrary often differentiable

I know that :) Cauchy Integral formula^

right :) having a holomorphic function is a great thing :)

Ok, I then will check the thing for myself that the CR equations also give the single points where $f'$ exists

I just wanted some clarity about that because its an test-exam and I just knew stuff about open subsets etc.

ok good :)

thanks for your time :)

you are welcome