The exercise:
Let $\mathbb{D}$ be the unit disk on $\mathbb{C}$. Let $f:\overline{\mathbb{D}}^2\rightarrow \mathbb{C}$ be a continuous function, holomorphic in $\mathbb{D}^2=\mathbb{D}\times \mathbb{D}$. Fix $\theta \in \mathbb{R}$. Prove that $$ g(\zeta) = f(\zeta, e^{i\theta}) $$ is holomorphic on $\mathbb{D}$.
What I've tried:
My attempt consisted of checkin that $g(\zeta)$ verified Cauchy-Riemann's equations, that is, if $\zeta=x+iy$, then I should have $$ \dfrac{\partial g}{\partial x} = -\dfrac{\partial g}{\partial y}. $$
Using the limit definition, for $\delta$ real: $$ \begin{array}{rcl} \lim_{\delta\rightarrow 0} \dfrac{g(\zeta + \delta) - g(\zeta)}{\delta} &=& \lim_{\delta\rightarrow 0} \dfrac{f(\zeta + \delta, e^{i\theta}) - f(\zeta,e^{i\theta})}{\delta} \hspace{15pt}(\text{ from the definition })\\ &=&\lim_{\delta\rightarrow 0} \lim_{r\rightarrow 1^-} \dfrac{f(\zeta + \delta, re^{i\theta}) - f(\zeta,re^{i\theta})}{\delta} \hspace{15pt}(\text{ from continuity })\\ &=&\lim_{r\rightarrow 1^-}\lim_{\delta\rightarrow 0} \dfrac{f(\zeta + \delta, re^{i\theta}) - f(\zeta,re^{i\theta})}{\delta} \hspace{15pt}(r<1 \text{ means functions are holomorphic})\\ &=&\lim_{r\rightarrow 1^-} \dfrac{\partial f}{\partial x}(\zeta,re^{i\theta}) \\ &=& \lim_{r\rightarrow 1^-} -\dfrac{\partial f}{\partial y}(\zeta,re^{i\theta}) \hspace{15pt}\text{ (Cauchy-Riemann's equations on 1st variable)} \\ &=&\lim_{r\rightarrow 1^-}\lim_{\delta\rightarrow 0} -\left(\dfrac{f(\zeta + i\delta, re^{i\theta}) - f(\zeta,re^{i\theta})}{i\delta}\right) \\ &=&\lim_{\delta\rightarrow 0}\lim_{r\rightarrow 1^-} -\left(\dfrac{f(\zeta + i\delta, re^{i\theta}) - f(\zeta,re^{i\theta})}{i\delta}\right) \\ &=& -\dfrac{\partial g}{\partial y}. \end{array} $$
One thing I must verify is that I can switch the order of the limits on the above calculations. My attempt was to get any sequence $\{ r_n\}_{n=1}^\infty$, $0 <r_n <1, \forall n$ and $r_n\rightarrow 1^-$ and consider the sequence of functions $f_n(\zeta,r_ne^{i\theta})$ and prove that $f_n \rightarrow f$ uniformly. The commutativity of limits then follow from Moore-Osgood's Theorem. This is where I'm stuck. I'm trying to work with power series on the interior of the domain to no avail
Would someone tell me if this attempt is ok? Maybe give me some help?
BTW, in case you need the reference from this, it's from Jiri Lebl's freely available book: https://www.jirka.org/scv/ it's exercise 1.1.1.