A book I'm reading claims that $\overline{Exp(z)}= Exp(\overline z)$ where the exponetial function is defined as $\sum_ {n=0 }^{\infty } \frac{z^n } {n! } $
Verifying this would mean verifying that $\overline {\lim_ {k \to \infty } \sum_{n=0 } ^{k } \frac {z^n } {n! } }= \lim _ {k \to \infty }\overline {\sum _{n=0 } ^{k }\frac {z^n } {n! } } $, but I don't know about any rules for taking conjugation inside a limit. How is this motivated?
Note that I don't think this is a duplicate of Why is $ \overline{e^z} = e^\overline{z} $? since Gedgars answer does not motivate why this can be done.
Complex conjugation is continuous.
To prove this, first prove $$ \overline{z-w} = \overline{z} - \overline{w} $$ then use it to show $$ \bigl|\overline{z}-\overline{w}\bigr| = \bigl|z-w\bigr| $$ Finally, in the $\varepsilon$-$\delta$ definition of "continuous", take $\varepsilon = \delta$.