In complex analysis differentiability for a function ##f## at a point ##z_0## in the interior of the domain of ##f## is defined as the existence of the limit(adsbygoogle = window.adsbygoogle || []).push({});

$$ \lim_{h\rightarrow{}0}\frac{f(z_0+h)-f(z_0)}{h}.$$

But why are the possible ##z_0##'s in the closure of the domain of the original function? And what are the possible "candidates" for this limit?

I know that for a function ##f:A\rightarrow{}B##, where the domain ##A## is a subset of some topological space ##X## and the image ##B## is a subset of some other topological space ##Y##, the limit as ##x## approaches ##x_0## of ##f(x)## equals some ##L##, i.e.

$$ \lim_{x\rightarrow{}x_0}f(x)=L, $$

if and only if for all neighborhoods ##V## of ##L## there exists a neighborhood ##U## of ##x_0## such that ##f(U\cap{}A-\{x_0\})\subseteq{}V\cap{}B##.

In this case ##x_0## is required to be a limit point of ##A## and ##L## in the closure of ##B##.

So are the allowed ##z_0##'s in the expression above limit points of something (i.e. is the interior of the domain of the original function equal to the set of limit points of something)? And are the allowed values of the "differentiability limit" in the closure of something else?

I know the definition of limits for a function only requires that both the domain and the image be subsets of some topological spaces, which means that I could define limits for complex-valued functions of a complex variable just from the topological structure of ##\mathbb{C}##. However, differentiability requires more structure, right? From the reverse perspective, differentiability in complex analysis involves someoperations, which are not part of the topological structure of ##\mathbb{C}##. So does my question about the allowed values for the "differentiability limit" even make sense?

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# Allowed values for the "differentiability limit" in complex analysis

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