$F:G\backslash \{d \}\rightarrow \mathbb{C}$ holomorphic , F' holomorphically extendable over d. It will now be shown that F is then also holomorphically extendable over d.
Assumption: $F:G\backslash\{d\}\rightarrow \mathbb{C}$ is holomorphic, but does not have a removable singularity at d, so it is not holomorphically extendable over d.
Let h(z)= \begin{cases}(z-d)^{2}F'(z), & z\ne d,\\ z=0, & z=d. \end{cases}
Then :$h'(d) = \lim _{z\rightarrow d} \frac{(z-d)^2 F'(z)}{(z-d)} = 0$
Is not true, but then there can't be a taylor series of $h(z)$ about d and so F'(z) could not be holomorphically extendable over d.
So from this it follows that if F'(z) has a removable singularity at d, so does F(z).
Can anybody tell me if this is correct?